My opponent will then proceed to exile their entire library, but once their library is empty, Helm of Obedience does not stop resolving and in fact will never stop according to the card text.
I argue that an infinite loop with only mandatory actions is created, and the game ends in a draw according to rule 421.4:
421.4. If the loop contains only mandatory actions, the game ends in a draw. (See rule 102.4b.)
Some people on the internet argue that instead, the effect of Helm of Obedience ends when there are no cards left to put in the graveyard because the effect cannot resolve, but rulings on HoO say
10/1/2008 If an effect like that of Leyline of the Void prevents cards from being put into your opponent's graveyard, the process described in the first sentence of Helm of Obedience's effect will never stop. Your opponent's entire library will be exiled, even if X is 1.
The phrasing "never stops" should indicate that the effect is, in fact, infinite, but the ruling does not actually cover the case when there are no cards to exile left, simply that this interaction is as described in this post, so which is it?