Argthtjtr

Obviously the difference between these two sentences is that one is using an adverb while the other one is using an adjective.

The reason why I think that an adjective should be used, is that the adjective refers to the object. By my understanding, if I take someone serious, I take them as a serious person.

When using an adverb, on the other hand, it refers to the verb, so whatever I am doing (in this case "taking someone") is being done in a serious way.

I can't take him serious. -> I am not able to take him as being
serious, or as a serious person.

I can't take him seriously. -> I really am unable to take him, or I
can't take him while being serious myself.

I think in this case "to take someone (serious)" is equivalent to "to consider someone (serious)". If instead I said I can't consider him seriously, I think the sentence would be missing an adjective, and also the adverb would be in the wrong place, e.g. I seriously can't consider him a serious person.

So in conclusion I think the sentence should be either:

I can't take him serious.

or using the adverb in front of the verb, e.g.:

I seriously can't take him anymore.

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.

One of the important features of English which allows non-trivial sentences to be "disambiguated" and understood is that adjectives and their matching adverbs are usually spelled/pronounced differently. This means that whether the word modifies the nearest noun or instead the nearest verb ("nearest" here meaning syntactically closest) can be determined by the form of the word. (I suspect many languages have this feature, but it's not obvious to the "fish in water".)

What that means in the case of:

I can't take him serious

is that "serious", being an adjective, must modify "him". Whereas:

I can't take him seriously

has "seriously" modifying "take".

In the usual sense of the above idiom, the meaning being conveyed is that "he" is hard to believe (perhaps a liar), or perhaps he's always joking. For this meaning you want to modify "take", since it's your perception that is being described, not "him" per se.

If you say "I can't take him serious" you have something of an incomplete/nonsensical statement. "I can't take him" is something you might say if you were told he needed a ride somewhere, eg, but tacking on "serious" seems to imply that the reason you can't give him that ride is that he's too serious.