Learn Russian: Откройте рот, пожалуйста.

In Russian, it's very common to leave out possessive words like 'my' or 'your' when talking about body parts, clothing, or things that obviously belong to the person doing the action. Saying ваш рот here would sound unnatural.

Why does the verb end in -те (Откройте)?

The -те ending shows that this is the formal or plural command form (imperative). You would say Откройте to a patient, a customer, or someone you call вы (formal 'you'). If you were talking to a child or a close friend, you would use the informal ты form: Открой.

I know рот is the direct object here. Why doesn't its ending change?

You are right that рот is the direct object, meaning it is in the accusative case. However, рот is a masculine, inanimate noun. For masculine inanimate nouns, the accusative case looks exactly the same as the dictionary form (nominative case).

Is Откройте perfective or imperfective, and why?

It is perfective (the infinitive is открыть). In Russian, polite requests to do a single, completed action right now usually use the perfective aspect.