What's the corresponding term for the mouth?

Solution 1:

I think the figure of speech you're looking for is to hold your tongue.

hold one's tongue (TFD)

Fig. to refrain from speaking; to refrain from saying something unpleasant. I felt like scolding her, but I held my tongue. Hold your tongue, John. You can't talk to me that way!

Solution 2:

Metaphors for keeping quiet are generally transitive, for example muzzled or gagged.

The protestors were effectively gagged by the court order.
Opposition leaders accused him of muzzling the news media [example from ODO]

The reason for this is probably because it's something one would not normally choose to do to oneself, unlike wearing earplugs for example.

Synonyms for gag listed by ODO include mute, which can also be an adjective and might be applied to people unable to speak through disability — so it's not necessarily the result of a transitive verb.

He was mute about the changes to be made.