Short answer: no.
There is much debate on this topic and it largely depends on the store itself and your relationship to the folks there.
Having said that, unless the VZW employee specifically looks for root access, they really won't know (the about page doesn't change). Even if they did know, would it be relevant and would they care?
Flashing stock ROMS or even some, at least, custom ROMs doesn't increase the counter. I don't know what actually increments it.
Personally, if my phone were in such sad shape that I would actually bring it to the store, I would have put bone stock ROM back onto it for troubleshooting purposes anyway and to quantify that, indeed, there's a problem I did not cause and cannot fix.