I have a 1999 Yamaha Big Bear 350 and I have noticed gear oil leaking from the seal where the axle enters the differential. I purchased a new oil seal from the dealer, but I can't seem to remove the axle so that I can change out the bad one. Does anyone know how to remove the axle on this machine? I have spoken to some friends and they say the axle should slide right out, but this is not the case with this particular machine.  

well lets look at the problem closer.

sometimes when the seal goes out it's because the axle bearings are going out or have gone out.
causing play in the axle which causes the seal to leak.

jack the atv up and grab both wheels and see if the axle moves front to back in the axle housing.
(push one wheel forward and pull the other wheel back)

when the axle bearings go out, one or more of the bearing races will heat into the axle housing and the axle wont come out without some persuasion with a hammer. it sounds like this is most likely the problem.