Well, here is what happened. I sanded a few spots where the paint was peeling. I then mopped the floor without any cleaner to get off the drywall and sawdust. As soon as the water hit the sanded spots, the floor paint started to bubble.
So, I dumped water on the entire area, and then used a hand paint scraper to remove every last bit of paint. Not fun, but I hope it was worth it. Tile is down, hands and knees hurt.