Learn Portuguese (Portugal): Não interrompas a professora!

Because this is a negative command for tu (informal you). In Portuguese, telling someone NOT to do something uses a different verb form called the present subjunctive. For '-er' verbs like interromper, the normal '-es' ending changes to '-as', making it não interrompas.

How would I say this formally if I were talking to someone using você?

You would say Não interrompa a professora! For formal commands, you use the você form of the present subjunctive, which drops the 's' at the end.

Is a professora the person being spoken to, or the person being interrupted?

It is the person being interrupted (the direct object). You are telling 'you' (the listener) not to interrupt the teacher. If you were talking TO the teacher, you would use a comma: Não interrompas, professora! (meaning 'Don't interrupt, teacher!').

What if I wanted to say 'Do not interrupt me!' instead?

You would say Não me interrompas! In negative sentences, words like não act as magnets and pull the object pronoun (me) to be placed right before the verb.