Learn Russian: Нет, мы пьём только сок.

Пить (to drink) is a common irregular verb. Instead of just dropping the -ть and adding regular endings, the base of the word changes in the present tense. It takes a soft sign (ь) and uses endings with ё. So for мы (we), it becomes мы пьём.

Do the other present tense forms of пить also use the soft sign and ё?

Yes! Except for the я (I) and они (they) forms, all others use -ьё-. For example, он пьёт (he drinks) and вы пьёте (you drink). The я form is я пью and the они form is они пьют. Also, remember that the letter ё is always the stressed syllable!

Сок (juice) is the thing being drunk. Shouldn't its ending change because it is the direct object?

It is in the direct object case (the accusative case), but you don't see a spelling change! For masculine nouns that are non-living objects (inanimate), the accusative form looks exactly the same as the dictionary form. That is why сок does not change here.

Why is только (only) placed right before сок?