In the interest of building a decent article, here's what I have gleaned from a little internet sleuthing.
The idea is to get pesticides into the canopy, and under the trunk-wood, but not in the sapwood (as I suggested in a comment). This breaks the lifecycle and also gets the larva where they do the most damage.
Some of the soil, trunk-injection, and trunk-spray based treatments for Emerald Ash Borer contain pesticides like imidacloprid and clothianidin. At least one of these are common in the cocktails used for making pressure treated lumber.
This would suggest that using such lumber for indoor use might not be recommended. However, current treatment concentrates on the trunk and canopy, and even soil-based treatment should not enter the heartwood. And none of the treatments are remotely like PTL, where lumber has the chemical cocktail driven deep into the heartwood. In the case of an ash tree, once you strip off the trunkwood and any damage there made by the pest, it looks like you have reasonably safe lumber.
So, this is my weak conjecture that previously treated ash can be safely used for lumber in the usual way, at least for furniture and so on. Maybe I'd refrain from using it for bowls or other food use items simply out of an excess of safety; there might be better lumber you can use for that. And I'm probably not going to run out and carve wooden blocks and toys out of this stuff for babies.
I'm hand-waving over the recommendations for how treated ash timber and fresh sawn lumber is transported and stored, as there is evidence that such material is a vector for passing the pests on to other areas. So, follow your local laws about that. Also, take the usual care one should take when sawing any lumber up; even the cleanest organic lumber is toxic when turned into a fine, breathable dust.