It is not an adverb. It is like an adjective - but tricky as it is the passive voice. Here is a detailed answer online: https://www.quora.com/Since-the-word-alive-is-not-an-adverb-why-can-we-say-He-is-buried-alive

They're both modify nouns (He, man) so they're both adjectives. I can see why the first sentence could be a little confusing since "alive" follows the verb, however, "alive" does not describe the act of burning him, it describes his state when he was being burned.