My impulse is that it’s modifying the verb leads, and is thus an adverb; yet it seems that a case could also be made that it’s exerting power on the phrase to extinguishment, a noun, which would make it … what, exactly? an adjective? I’m confused.
Fame lights a fuse that leads only to extinguishment.
In that sentence, only is an adverb, but to extinguishment isn’t a noun. It’s a prepositional phrase. It’s quite common for adverbs to modify prepositions (e.g., completely under the bed). It’s also quite common for adverbs to modify entire noun phrases, (e.g., even a Möbius strip, only the lonely). They rarely modify individual nouns inside noun phrases, though they do occasionally.