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.