While looking at different Wikipedia articles on different X-Men characters, I found a character named Masque, who can reshape a person's skin any way he wishes. I was just curious: Could he do this and reshape their face so the holes to their nose and mouth were blocked?
Yes, he can. Masque's powers allow for fairly extensive reconfigurations. In addition to being able to alter hair and skin color, he can also remove orifices (and presumably add new ones).
Issue #179 contains both of these, used on Kitty Pryde:
In other encounters with the X-Men, Masque has also closed up Banshee's mouth so he couldn't speak and created tentacle arms for Callisto.