"young" in "to die young" is a normal adjective.The way someone can die can be peacefully or painfully, but the way of dying can't be young. "young" stands for "to die in young age" or "when someone is young". One does not think of the manner of dying but of the age of someone who died.
The school rule that after a verb follows an adverb is an oversimplification. There are a lot of verbs followed by an adjective where we clearly don't think of manner but of something else. But, of course, for beginners it is difficult to feel the difference and the grammar point verb + adjective is not a point in course books. It is advanced grammar.
Some examples
to buy sth cheap (at a cheap price, not in a cheap manner)
to break free - He tried to break free (to get free)
to fly high - (at a high level or to a high point)
to ring true (that rings like sth that is true)
3I don't see any copula or adverb here. Young is an adjective functioning as a depictive secondary predicate. In fact, this is pretty much a textbook example. Someone should make a canonical question about this stuff since it comes up so often, and people keep calling them adverbs. – snailplane – 2015-05-06T08:25:20.207
I see a complement in "young". – IllidanS4 wants Monica back – 2015-05-06T09:39:58.567
1Grammatical how? As a complete sentence? Or as a clause? – J.R. – 2015-05-06T15:13:11.010
@snailboat Looked up secondary predicate. That also sounds like a reasonable interpretation. Is there any reason 'die' couldn't be acting as a copula, with 'young' as a secondary predicate? I'm really not that familiar with this grammatical structure in a formal sense.
– DCShannon – 2015-05-06T18:44:29.363@J.R. Good point. I don't really have trouble with sentence, clause, phrase or sentence fragment. Question updated. – NS.X. – 2015-05-06T18:59:05.283