Yes, he should hunt with his .25-06 if:
the .25-06 is all he has,
there is no one he can borrow a more fitting rifle from,
his funds don't allow him to purchase something more fitting,
he used a bullet that will not shed ANY weight (Barnes or other all-copper),
he is 100% sure he will put the bullet behind the shoulder muscle into the lungs,
he hunts where he can recover an elk even if it doesn't drop soon,
he's willing to forgo any shot beyond 300 yards if there is a rest or any shot beyond 200 yards without a rest,
and (most importantly),
he will forgo any shot, even at a MONSTER elk, that isn't 100% positive to be within these constraints.
A huge part of this discussion wasn't given to us, "where he's hunting". If he's hunting the remote mountain wilderness with lots of timber, a .25-06 is a very poor choice since he may never find the elk he shot. In open terrain where recovery of game is much easier, a slow-kill should still mean a recovery so a .25-06 isn't as bad of an idea.
Me personally, I would find something more substantial to kill an elk with as I know I would not want to pass on a 8x8 bull because he wouldn't give me a quartering away shot within 300 yards! As hard as it is for me to get elk hunting, I don't want to be hunting with a rifle that may mean a failed hunt.
People have killed elk with smaller rounds. It can be done, but a .25-06 still leaves you with virtually no margin for error in a game where everything doesn't always go as planned.
