Unfortunately we don't have this level of detail about urushiol and how quickly it degrades once spread in the environment. Different references say different things, ranging from a year to several years when residues contaminate tools or fabrics, and that time frame may depend on conditions like exposure to UV light, microbes in the environment, and so forth. On tools where contamination is known, like mower blades, the best approach is probably just to wash contaminated surfaces off with a degreasing detergent and water. As the grass grows, any oils left on the grass blades will eventually get cut off and will decompose on the soil surface (if you use a mulching mower that doesn't bag the clippings, at least), so over time, we'd expect the lawn will outgrow any residues.

As for protecting the dog from exposure, we don't know...you'd have to ask a vet about what level of risk may be acceptable based on the dog's size, how likely they are to encounter the most heavily-contaminated parts of the yard (closest to where the mower first ran over the poison ivy, since it will presumably be more dilute the further away you go), and behaviorally, how to discourage them from sniffing or spending time in an area you are most concerned about.

Miri

Thank you, Miri!!!!

You're welcome!