Should we tile under the water heater? If not, will a drip pan be ok sitting on subfloor?

I'd drain and remove the water heater but I'm sure there are plenty of hacks here that wouldn't.

How old is the water heater? electric or gas? If it's gas and 15+ years old, I'd probably replace it. Fifteen years is about the typical lifespan, maybe less. If that isn't a factor, then do you care how it looks? If you can do it yourself, it's not much work to take it out and put it back and I'd do that. Another alternative would be to leave the one area without tile, but save tile and grout so that it can be done later when the WH is replaced. But again, unless it's DIY, calling someone back to do the tile might be as much as getting a handyman type to remove and replace the WH right now.