Here's my guess: Possibly a little bit, but not as much as you might think -- I am assuming that you don't hear an obvious problem with them, since you wouldn't be asking if that were the case. Despite what would seem intuitive, the dome shape doesn't have much effect on dispersion -- that's determined by size, not shape. It might have some effect on resonances (whether dents would cause a problem or, instead, an improvement, is less clear than you might think, since a mushed dome might have resonances that are spread out, which might be better than the more defined resonances of the dome, and the softer mushed paper might be more self damping.) And the dome is probably only really important at the highest frequencies, which you may be unable to hear.
Cosmetically, the pushing in is unfortunate, and hard to repair in a way that doesn't still look bad, but you could try if it bothers you. Because its paper, pulling with tape or with a vacuum -- careful not to pull the entire dome off -- isn't likely to work. But using a pin poked through and then pulling/levering up the low parts will work more or less well. A bit of glue will seal the hole.