Is tense an adverb

No, tense can be an adjective, noun, or verb, but not an adverb. However tensely is an adverb.

tense (adjective) - taut, under tension; or anxious, nervous, worried (when used for a person)

tense (noun) - grammatical verb conjugation, primarily by time (e.g. past tense)

tense (verb) - to stiffen (as in preparation for a kinetic action), or to flex a muscle.