Why is it "to take someone seriously" and not "to take someone serious"?

Alas, grammar can take you only so far. Idiom takes over here, and idiom directs that "Can't take him X" requires that X be an adverb modifying "take." In fact, it is almost always the case that X="seriously", so "take him seriously" is almost a set phrase. To see what I mean, try using the google or the Ngram viewer with X="honestly" or "facetiously" or "gratefully.

Thus your first solution ("I can't take him serious.") fights idiomatic usage. Your second solution ("I seriously can't take him.") means something else, namely that you really can't put up with him.

Other verbs don't have the same idiomatic restriction. Consider

I can't consider him seriously.
I can't consider him serious.

The first means that you don't think he's a viable candidate for whatever position you're thinking about; the second means that you don't think he's a serious person. Or

I can't remember him angrily.
I can't remember him angry.

The first means that you can't think back about him with anger; the second means that you can't think of a time when he was angry.

Read this sitting down. Take this with a grain of salt.

This is not sitting down and this doesn't have its own grain of salt. The complement is a verbal complement rather than a modifier of the direct object. So, with

Take him seriously.

seriously is how you take him.

Take serious is THE VERB. It's not TAKE as an auxiliary verb plus an adverb. That's why TAKE SO OR STH SERIOUSLY sounds wrong to me. SERIOUSLY is not the way you TAKE so or sth, when you take so/sth serious. Only correct phrase using the adverb: I can't take him, seriously