Learn English – Grammatically correct: ‘are you hurt’ and ‘do you hurt’

Building on Joe's answer, grammatically, the "hurt" in

"Are you hurt?"

is a past participle of the verb "to hurt", used formally in the transitive sense (= to cause pain or injury to someone), but combined here with the auxiliary "be" to form passive voice.

On the other hand, the "hurt" in

"Do you hurt?"

is just the bare infinitive of the intransitive verb "to hurt" (= to be in pain), used with the auxiliary "do" to form a present tense interrogative.

The verb "hurt" just happens to be irregular, such that its past participle looks identical to the bare infinitive. The grammar becomes clearer if we replace it with more regular verbs like, say, "injure" and "ache" (which also have the advantage that "injure" is always a transitive verb, whereas "ache" is generally intransitive):

"Are you injured?"   (be + past participle)
"Do you ache?"   (do + bare infinitive)

And yes, the past participle of English verbs behaves grammatically like an adjective, so it's also legitimate to analyze the first sentence as "Are you <adjective>?"

Indeed, several English adjectives (including, notably, the word "past" itself) started out as past participles of verbs, but later lost the connection with the original verb. This can happen because the verb itself has otherwise fallen out of use (for example, "crook" is rarely used as a verb any more, but "crooked" is still a perfectly good adjective) or because the original irregular participle has been replaced by a more regular one (as in "past" vs. "passed", or "wrought" vs. "worked").

Even when both the verb and its participle remain in common use, it's not unusual for the participle to acquire secondary meanings or connotations that don't directly map back to the verb. For example, off the top of my head, the adjective "baked", while clearly a regular past participle of the verb "bake", also has the slang meaning "high on marijuana" that doesn't correspond with any meaning of the original verb.

Arguably, this is somewhat true of "hurt", too — even though it's originally a past participle of a transitive verb, in common usage it doesn't really carry any connotation of there being an active agent that caused the injury. That is, you can be hurt (adjective = past participle) even if nobody has hurt (past tense) you.