Learn German: Ich muss mir das Gesicht waschen.

Because there is already a direct object in the sentence: das Gesicht (the face). When you do an action to yourself and name a specific object receiving the action, the reflexive pronoun becomes the indirect object and takes the dative case (mir).

Why does it say das Gesicht (the face) instead of mein Gesicht (my face)?

In English, we say 'wash my face', but in German, it is standard to use a definite article (like das) for body parts when the owner is already clear. The word mir already tells us whose face is being washed.

If I didn't mention my face, would I still use mir?

No! If you just wanted to say 'I have to wash myself', you would use the accusative pronoun and say Ich muss mich waschen. You only switch to the dative mir because you added the specific thing you are washing (das Gesicht).

Why is waschen at the very end of the sentence?

This is a quick reminder about how modal verbs work. The modal verb muss (must) takes the second position in the sentence and pushes the main action verb (waschen) to the very end in its infinitive (dictionary) form.