Can I tile over a painted concrete floor?

Yes, but you will most likely need to use a floor leveler.

NOPE. If the paint comes loose, so does the tile. You should either strip or bead blast the concrete and thoroughly clean it. Water should readily absorb in the concrete as an indication that it is suitable for bonding the tile to it. Floor leveling compound won't hold to the painted surface either.

https://www.ehow.com/how_5569050_install-tile-painted-concrete-floor.html

absolutely!

I had a concrete floor in my restaurant and very limited funds for such a large space. After some research I ended up cutting the concrete into 2x2 squares, staining it, and then grouted the cuts. It was a huge amount of work but the result was stunning and it held up really well (would have been better if I had waxed or sealed it but I got tired lol). The cutting wasn’t necessary, just my preference at the time, and I think it would have looked just as amazing if I hadn’t with much less work and mess.