Learn Russian: Не нужно беспокоить директора, он занят. - No need to disturb the director, he is busy.

When expressing that someone doesn't need to do something or shouldn't do it using не надо or не нужно, Russian almost always uses the imperfective infinitive. Even if you mean "don't disturb him this one specific time", the negative obligation triggers the imperfective aspect to say "avoid this action altogether".

Why does директора end in -а here? Is that the genitive case?

It looks identical to the genitive, but it is actually the accusative case because the director is the direct object of the verb. For animate masculine nouns (living beings like a director), the accusative case takes the exact same endings as the genitive case.

Could we use не надо instead of не нужно in this sentence?

Yes, they are interchangeable here. Both mean "there is no need" or "one shouldn't". Не надо is extremely common in everyday speech, while не нужно can sound just a little bit softer or more formal.

Why does the sentence use занят instead of the full adjective занятый?

Занят is a short-form adjective. Russian frequently uses short-form adjectives to describe a temporary state (he is busy right now) that acts as the main predicate of the sentence. You would use the full form занятый if it were directly attached to the noun, like занятый человек (a busy person).