What's the corresponding term for the mouth?

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.

Wearing a muzzle could be used metaphorically, but the meaning would not be clear. To me it conveys the idea of being forced not to talk. Choosing to keep silent can be expressed as:

to keep mum or keep one's mouth shut:

• refuse to talk or stop talking; fall silent; "The children shut up when their father approached"

also

to button up - stop talking; fall silent;

buttoned up: taciturn; silent and somewhat tense

One oft-used idiom for remaining silent is "keeping your own counsel" (Cambridge Dictionaries)

• to not say what your opinions are

It was explained that you wouldn't choose to muzzle yourself. I don't think you would put a damper on your speech either. But there is self-censorship when there is no external agency involved in the censoring. Whether figuratively speaking to self-censor would actually be about keeping silent when you oughtn't is another question altogether. It may not be very idiomatic but does showcase the choice the person makes and the muzzling effect.