Felt will damp down some high frequencies so that's why there is some stuck to the back of the tweeter, because it's a flat metal surface. But if you removed it then it's pretty unlikely that you'd notice any difference.
There's no hard rules with this sort of thing, Some advocate no damping at all. Some advocate stuffing the box full and then you've got all points in between. In my own experience designing and building some large-ish infinite baffles I tried the full range of options over the course of time and found no damping at all to give the best sound.
However the cab walls were pretty thick, 3 quarter inch all round. Many commercial speakers have cabs that are as light and flimsy as they can get away with so heavy damping becomes a necessary evil. In my opinion it reduces clarity and bass quality. But that's just my opinion/experience, plenty will disagree.
Current Lash Up:
TEAC VRDS 701T > Sony TAE1000ESD > Krell KSA50S > JM Labs Focal Electra 926.