For that matter, a physical digital computer has a bounded number of states it can be in, so it can't be Turing complete at all, if you're going to be really pedantic.
You're only allowed to say MTGO is not Turing complete if you also agree that Java is not Turing complete (because the 32-bit JVM has bounded memory it can access).
There's also the infamous video of (well-known professional player) Luis Scott-Vargas "breaking" MTGO by causing the game to wind up in a non-terminating loop of mandatory actions:
Its effect is that, when it comes into play, you choose a non-land card in play that gets exiled (removed from play) for as long as Oblivion Ring stays in play, then gets returned once Oblivion Ring leaves play. This game devolves into a situation where there are three Oblivion Rings and no other non-land cards, so: Ring 1 exiles Ring 2, which returns Ring 3, which exiles Ring 1, which returns Ring 2, which exiles Ring 3, which returns Ring 1, which exiles Ring 2, and so on to infinity. The rules state that when a loop like this occurs (which is not often in real play), the game is an automatic draw.
You're only allowed to say MTGO is not Turing complete if you also agree that Java is not Turing complete (because the 32-bit JVM has bounded memory it can access).