What is the shelf life of cookies made with Greek Yogurt?

In general, the yogurt is not going to affect the shelf life of any baked goods. You're going to kill the culture when you bake it, so it's just a question of how long that particular baked good lasts.

And so the answer varies drastically depending on what you've made. It's mostly a function of how moist it is. Dry cookies last several days or even weeks at room temperature, just slowly getting stale. Don't bother putting them in the fridge; it'll just make them go stale faster. Very moist cookies might start to mold at room temperature, so they're best eaten quickly. Pies and cakes generally need to be kept in the fridge. They'll generally last at least several days, with cakes tending to dry out and pie crust suffering a bit.

If you want things to last longer, freeze them. Everything essentially lasts forever in the freezer: it's safe forever and all you have to worry about is decline in quality. You just need to make sure they're stored airtight with as little extra space as possible so that they don't take on smells and they don't accumulate a lot of frost.