If it is the mesh over the front of the speaker - no it won't, you can take it all off with no problem.
Doesn't look quite as smart but will sound the same.
Or do you mean the rubbery membrane that supports the cone to the speaker chassis is starting to perish ?
If so yes eventually it will but unless youre a real audiophile you may not notice for quite some time.
Like rubber bands they dry out and crumble away
If it is the rubbery surround then smear it lightly with petroleum jelly (both sides if possible) to try and keep whats left of it intact and reduce the loss of moisture in the rubber.
If you push a lot of watts through them then the cone will move more and further and will break the rubber even quicker, so try and listen at lower volumes to stop the cone moving so much.
sooner or later though you will need to replace them
The material is between the rubber and the cone about 1" wide. It has left a gap where it has eroded. Not sure whether it will have any long term effects on the speakers.
"Not sure whether it will have any long term effects on the speakers."
Yes, it most definitely will.
The purpose of the surround is to keep the cone centred. If the surround is damaged, the cone can shift, and the spider (which is the part that centres the cone at the magnet end) will act as a fulcrum, the cone can now pivot on the spider,causing the voice-coil to rub on the sides of the gap.