I cannot give you a definitive answer. The toxin could exist for a long period of time but again gave no definite time. This Web site has this to say:
wizard.pharm.wayne.edu...

"The urushiol may be active for months on tools, sports equipment, shoes, and clothing, especially in a dry atmosphere. Stroking a pet whose fur is contaminated is also a common cause of allergic reaction."
Here is a write-up on poison oak but I do not find a reference about how long the oil is viable. All members of the genus Toxicodendron, including poison oak, poison ivy, and poison sumac, cause allergic contact dermatitis. About two million cases of skin poisoning are reported in the United States each year, primarily caused by these three species. In California, the number of working hours lost as a result of dermatitis caused by poison oak make it the most hazardous plant in the state. Contact with poison oak leaves or stems at any time of the year can cause an allergic response. When the allergen contacts the skin surface in sensitive individuals, it is rapidly absorbed into the surrounding cells. Within 1 to 6 days, skin irritation and itching will be followed by water blisters, which can exude serum. Contrary to popular belief, the exuded serum does not contain the allergen and cannot be transmitted to other regions of the body or other individuals. The dermatitis rarely lasts more than 10 days.

Although 50% of the population is clinically sensitive to poison oak and poison ivy, about 75 to 85% can potentially develop allergy if exposed to a sufficiently high concentration of the toxin. Once a reaction to the toxin has occurred, the body responds with a cell-mediated immunity, which is a delayed hypersensitivity. Those individuals who have developed delayed hypersensitivity are sensitive to the toxin and repeated exposures further increase sensitivity. Conversely, long periods with no exposure will reduce an individual's susceptibility to the allergen.

There is no known difference in sensitivity to poison oak among races or between sexes. Animals do not usually suffer skin irritation from contact with poison oak because they are protected by fur; dogs, however, can contact poison oak on their nose or underbelly. On the other hand, livestock may graze on the tender foliage. In addition to direct contact with the plant, transmission of the allergen can occur from a number of other sources including smoke particles, contact with objects such as clothing, gloves, and tools, or contact with animals, particularly pets. When poison oak is burned, the oils can be transported on the smoke particles. Breathing this smoke can cause severe respiratory irritation. There's always the potential of transferring some of the oil present in the leaves, stem, fruits, roots or flowers of the poison oak plant to the skin. It is this poisonous oil which causes the skin irritation identified with poison oak. Although contact with the plant is normally the method of poisoning, an individual can also be poisoned by handling clothing, tools, objects or animals which have become contaminated with the oil. After coming in contact with the allergen, the best way to prevent skin irritation is to pour a mild solvent, such as isopropyl alcohol (rubbing alcohol), over the exposed area and then follow this with plenty of cold water (warm water enhances penetration of the oil) within a few minutes of exposure. If isopropyl alcohol is not available, just wash with lots of cold water. But you need to wash within 5 minutes of exposure to prevent a rash. Even if it is too late to prevent the rash, washing the skin to remove excess plant oil will keep the rash from spreading. The poison oak toxin is an oil, so it does not dissolve in water. Sufficient quantities of water, however, will dilute the oil to the point where it is no longer harmful.

Using only a small amount of water or disposable hand wipes is more likely to spread the toxin than remove it. Soaps can be used to wash, but only if with copious amounts of water; otherwise, they too will spread the toxin. An important point to remember when washing with isopropyl alcohol or soaps following exposure to poison oak is that they will remove the skin's protective oils. These oils help the skin to repel the plant toxin and will not regenerate for 3 to 6 hours following washing with these solvents. Therefore, wash with them only if you are done working outside for the day. If there is a possibility of re-exposure to poison oak within 6 hours, just wash with lots of water. Be sure to thoroughly wash the hands as they serve as the major route for transfer of the allergen to other parts of the body, especially the face.
BIOLOGY
Initial establishment of poison oak is generally by seed that is transported by birds. The single-seeded fruit are eaten by a variety of birds. The passage of the hard-seeded fruit through the bird's digestive tract facilitates germination by reducing the period of dormancy. Birds serve to disperse the seeds to new locations. Once established, the plant spreads by slow vegetative growth of underground horizontal rootstalks (actually stem tissue). A single root system can cover a very large area. Poison oak can survive under a wide range of temperatures, elevations, soil types, moisture conditions, and light intensities. However, it is most commonly found on hillsides with shallow soils. The flowers of poison oak are usually pollinated by insects.
MANAGEMENT
The primary ways of managing poison oak are mechanical removal by hand pulling (not recommended for individuals who are sensitive to poison oak) and treatment with herbicides. Maintaining a healthy cover of desirable vegetation will reduce the potential for invasion of poison oak. This is easiest where irrigation is available and the soil is regularly cultivated. Poison oak is a native species with its natural control agents already present. Consequently, biocontrol is not an option for the control of poison oak. Burning is not recommended for the control of poison oak. It not only creates a serious health hazard, but does not effectively reduce infestations. Grazing by sheep and goats can be effective in small areas. Deer or horses will also graze poison oak when the foliage is young, before the plant flowers. Mechanical Hand pulling or mechanical grubbing (using a shovel, pick, etc.) can be used to physically remove plants located in a yard or near houses. Remove plants in early spring or late fall when the soil is moist and the rootstalks are easily dislodged. Grubbing when the soil is dry and hard will usually break off the stems, leaving the rootstalks to vigorously resprout. Detached and dried brush can still cause dermatitis, so bury or stack the plant material in an out-of-the-way location, or take it to a disposal site. Never burn poison oak, even after physical removal.

Ideally, persons engaged in hand pulling poison oak should have a high degree of immunity to the allergen. Whether the individual is sensitive or believed to be immune, they should wear appropriate protective clothing, including washable cotton gloves over plastic gloves, when handling the plants. Wash all clothing thoroughly, including shoes, after exposure. Other forms of mechanical control have not proven to be successful. Brushrakes and bulldozers often leave pieces of rootstalks that can readily resprout. In some cases, brush removal late in summer, when plants are experiencing moisture stress, can slow their ability to recover. Mowing has little effect in poison oak control, unless it is performed repeatedly (at least four times during the growing season). Within 2 months of germination seedlings have usually produced underground rootstalks large enough to recover from mowing damage. A single plowing is of no value and often serves to propagate the shrub. However, good seedbed preparation and planting cultivated crops for a year or more will control poison oak infestations. Chemical Control Herbicides used to control poison oak include glyphosate (Roundup, Ortho Kleenup, etc.) and the auxinic herbicides triclopyr (Garlon, Ortho Brush-B-Gon, etc.), 2,4-D (Weedone, Spurge & Oxalis Killer, etc.), and dicamba (Banvel, Spurge & Oxalis Killer, etc.). These herbicides can be applied as stump or basal applications, or as a foliar spray.

Glyphosate is one of the most effective herbicides for the control of poison oak. However, effective control depends upon proper timing of application. Apply glyphosate late in the growth cycle: after fruit has formed but before leaves lose their green color. In hand-held equipment, glyphosate can be applied as a 2% solution in water. (Products or spray mixtures containing less than 2% glyphosate may not effectively control poison oak.) It is important to note that glyphosate is a nonselective compound and will damage or kill other vegetation it contacts. Auxinic herbicides, such as triclopyr, 2,4-D, dicamba, and combinations of these herbicides are also used to control poison oak. The application timing with auxinic herbicides is somewhat different than for glyphosate: applications can be made earlier than with glyphosate, when plants are growing rapidly from spring to mid-summer. Triclopyr is the most effective auxinic herbicide for control of poison oak. It has a wider treatment window than glyphosate and is often gives more consistent control. Two formulations of triclopyr are available. Triclopyr amine is the least effective of the formulations and requires relatively high rates. Triclopyr ester or triclopyr ester plus 2,4-D ester give better herbicide absorption into the foliage and are more effective.

When 2,4-D is combined with dicamba, it provides much better control than if it is used alone in a 1% solution. Premixed combinations of these herbicides are available. Dicamba applied at 0.5% gives better long-term control of poison oak than 2,4-D.
Stump Application. Stump treatments are most effective during periods of active growth. Cut stems of poison oak 1 to 2 inches above the soil surface and immediately after cutting, treat the stump. A delay in treatment will result in poor control. Apply an herbicide such as glyphosate, triclopyr or combinations of triclopyr with 2,4-D (or 2,4-D and 2,4-DP) with a 1- to 2-inch wide paint brush or with a plastic squeeze bottle that has a spout cap. Treatment solutions should contain either undiluted glyphosate (use a product that contains at least 20% glyphosate) or triclopyr amine or a 20 to 30% triclopyr ester solution mixed with 70 to 80% oil (diesel, kerosene, or fuel oil). Be sure to completely cover all surfaces of the stumps with the herbicide until it runs down the base of the stubs. At completion of treatment, thoroughly rinse brush or bottle several times and dispose of the rinse on the ground near the shrubs that were treated. Spray any regrowth from cut stumps with a foliar spray when the leaves fully expand.

Basal Application. Basal bark applications can be made almost any time of the year, even after leaves have discolored or dropped. Apply triclopyr to basal regions of poison oak by backpack sprayers using a solid cone, flat fan, or a straight-stream spray nozzle. Thoroughly cover a 6- to 12-inch basal section of the stem, but not to the point of runoff.

Foliar Sprays. The effectiveness of herbicides applied to poison oak foliage depends on three factors: (1) proper growth stage at time of application; (2) spray-to-wet coverage; and (3) proper concentration. To achieve spray-to-wet coverage, all leaves and stems should be glistening following herbicide application. However, coverage should not be to the point of runoff. Foliar application of herbicides to poison oak is most effective after leaves are fully developed and when the plant is actively growing. This period is normally from April into June or July, when soil moisture is still adequate. The flowering stage is the optimum time to spray. Do not apply herbicides before plants begin growth in spring or after the leaves have begun to turn yellow or red in late summer or fall. One application of a herbicide usually does not completely control poison oak. Retreat when new sprouting leaves are fully expanded, generally when the plants are about 2 feet tall. Watch treated areas closely for at least a year and retreat as necessary.
References:
Epstein, W.L. and V.S. Byers. Poison Oak and Poison Ivy Dermatitis - Prevention and Treatment in Forest Service Work. USDA Forest Service Equip. Dev. Ctr. Pub. 8167 2803, Forest Service Equip. Dev. Ctr. Missoula, MT 59801
Hauser, S.C. 1996. Nature's Revenge: The Secrets of Poison Ivy, Poison Oak, and Poison Sumac, and their Remedies. Lyons & Burford, N.Y.