Lewis & Short, Gaffiot, Georges and Forcellini agree that it is an adjective. Oxford appears to be alone with its opinion that it is an adverb, and I wonder if the entry itself has anything to say about that. Aliquid alicui necesse est is a very common expression, and as Draconis has explained, a neuter adjective hardly seems out of place here.
Spīrā-re necesse est homin-ī
breathe-INF necessary be.3SG.PRES human-DAT.SG
Breathing is necessary for a human.
I would call it an adjective without a qualm. It's linked to a nominal (an infinitive verb) by a copula, and that's a syntactic context that adjectives appear in:
Errā-re humān-um est
err-INF human-N.SG.NOM be.3SG.PRES
To err is human.
I don't think it's productive to attempt to determine the part of speech of this word - it's neither, since it modifies neither a noun nor a verb. necesse/necessum est, opus est, ūsus est and oportet are all predicative expressions that are semantically indivisible, forming the predicate together. If you classify necesse as an adjective, you'd have to ...