Why is "for six years" not the adverbial complement in this sentence?

Solution 1:

The source of your example is https://www.thefreedictionary.com/Complements.htm

The source also explains what a complement is. It says:

Complements are words or groups of words that are necessary to complete the meaning of another part of the sentence.

and the article also distinguishes them from adjunct modifiers.

In the example, the adjunct modifier is "for six years" because "for six years" modifies "in Ireland"

You can say "I have been living in Ireland", but you cannot say either "I have been living" or "I have been living for 6 years." Thus "in Ireland" is the phrase that is necessary to complete the meaning. It is thus, the complement.

Solution 2:

I have been living in Ireland for six years.

"For six years" is a temporal adjunct (your adverbial). It modifies the verb phrase, not the noun "Ireland".

But the PP "in Ireland" is obligatory for this sense of "living", which means “reside”, and hence is not an adjunct but a complement.

Note that there is no such thing as an 'adverbial complement'. Adverbial (adjunct) and complement are separate distinct functions just as subject and object are.

Note also that complement is not a semantic term, but a syntactic one. It has nothing to with completing the meaning of a sentence, but whether the dependent (a clause or phrase) is licensed by an appropriate head, or is obligatory for one reason or another.