Disclaimer: I am not a professional contractor. I've had to research this stuff when looking at old houses before, though.
Are they about 8 or 9 inches square? I've seen ceiling tiles like that before in a basement that was finished sometime between the mid-50s and mid-70s. They nailed 1x2 furring strips to the bottom of the joists, and then used long staples to attach the tiles to the furring strips. I believe these could have also been installed over drywall, plaster, etc, using an adhesive like Liquid Nails or similar. If you take one down (but please don't do that yet), it's similar material to the 2'x4' drop ceiling panels you can get today, but a little thinner.
The problem is that some of these tiles contained the all-purpose miracle material, asbestos. If they're from the late 70s or before they might; if they're newer than the early 80s or so, they might not. The only way to know for sure is to (carefully) get a piece of the tile off the ceiling and get it tested.
If the tiles don't contain asbestos, then you can pretty much do what you want. If they look like fiberglass, it's a good idea to use a respirator, long sleeves, etc if you tear them out - just like dealing with fiberglass insulation. I would probably try to take one or two of them off - either one that's already loose, or one in a corner or closet if available - and see what the ceiling is above that. If it's drywall or plaster, you might think about demoing the part of the ceiling where the water leak was, just so you can be sure that it's all dried out and no funky things are growing in there.
If they do contain asbestos, life is harder. I know that when I had an asbestos popcorn ceiling in a previous residence, one of the legal options was just to paint over it without disturbing it, but I don't know if that applies to the tiles. It might also be an option to paint it (to seal in the asbestos), get a good studfinder and some really long eye screws, install the eye screws through the tile into the joists, and then hang a modern drop ceiling like you normally would (so it looks better). I don't know if you need the full bunny suit routine to install those eye screws or not.
More information here - I am not affiliated with this site: http://inspectapedia.com/hazmat/Asbestos_Ceiling_T...