Like some of the other posters above, I've had many battle maps that had dry erase, wet erase and permanant marker on them, and all refused to come off once they dried. I seriously doubt you'll be able to get the marker off without damaging the mat itself. Coca Cola was spilt on one of our mats, and it took off the marker when it was wiped off, but the marker had only been on the mat for an hour or so, it couldn't hurt to try I guess.
Could I make a suggestion?
Our group got tired of buying new battlemaps everytime they got ruined by marker(s), so we invested in a couple of 4'x4' clear 1/4 inch plexiglass sheets. On one side, we drew lines in permanant blank ink, and then use the other side to draw on with dry erase markers. We've been using them now for about two years, and even if the dry erase has been on there for a couple weeks, the dry erase cleaner solvent always cleans it up with no problem.
The only drawback is that each sheet was about $30 apiece, but for us it was worth it.