I was taught that more beautiful is a comparative adjective. But when I think about it, wouldn't more be an adverb and beautiful be an adjective?
The simplest analysis is that, yes, more is functioning as an adverb in modifying the adjective beautiful:
Forming the comparative of adjectives and adverbs, especially those of more than one syllable.
Example: ‘for them enthusiasm is more important than talent’
Regarding how to describe the two words together, BillJ comments that "more beautiful is an adjective phrase headed by the adjective "beautiful" with the comparative adverb "more" as modifier."
However, in his comment John Lawler notes that "since more … than is part of the comparative construction (and only occurs there), more is not so much an intensifying adverb as an auxiliary, particle, or complementizer, like the words marking other constructions" [bold emphasis is mine].