It'll probably inherit most of the moves along with having its usual Level 1 moves (which in Eevee's case is Helping Hand, Tackle, Growl and Tail Whip). I'm not quite sure how tutor moves are passed down, but I'm aware of the others. I know that Iron Tail was a TM in 4th Gen, but in 5th Gen it was a tutor move (Smack Down replaced Iron Tail's TM slot in that generation) so it's a different circumstance depending on generation there, but with the other three moves mentioned the Eevee will inherit those at once.
Also, if a possible combination of inherited moves causes all four of the baby's moveslots to be full when hatched, it won't learn any of the starting Level 1 moves unless if taken to the move relearner.
Look at this Bulbapedia page to see how the egg move inheritance system works:
http://bulbapedia.bulbagarden.net/wiki/Egg_move