Learn English – Can you say I don’t and can’t do anything

There is nothing wrong with saying

They don't and can't do chores.

This expression uses the rhetorical device known as zeugma. From Sylvae Rhetorica:

A general term describing when one part of speech (most often the main verb, but sometimes a noun) governs two or more other parts of a sentence (often in a series).
. . .
Zeugma comprises several more specialized terms, all of which employ ellipsis and parallelism (among the governed members of the sentence). The zeugma figures are of two types: those in which the governing word is the main verb (in which case these are subsequently categorized according to the position of that governing verb), and those in which the governing word is another part of speech (usually the subject noun).

In this case, you're linking two verbs to the same object. The do is elided from the first verb, making it kind of a double-jointed zeugma, in that it links both don't and can't to the helper verb do, and then to the object chores.

EDIT: To respond to an objection that "don't have to" is not the same as "don't" (a premise that is arguable if not as far-reaching as the objecter wishes to make it out to be) I will note that the zeugma works just as well with the wordier

They don't have to and can't do chores.

The point is, if I say I "don't do something" it can mean a number of things: that I am not required to do them, refuse to do them, have never had the occasion to do them, or any of a number of things. It's ambiguous, to be sure, but that is how people talk.