Learn German: Zeig mir deine Zunge!

The verb zeigen (to show) takes two objects: the thing being shown and the person it is shown to. In German, the person receiving the action is the indirect object and takes the dative case (mir), while the thing being shown is the direct object and takes the accusative case (deine Zunge).

Why is the verb just Zeig without any ending?

This is the informal singular command (imperative) form, used when talking to one person you address as du. To form it, you take the present tense du form (zeigst) and remove the -st ending, leaving just zeig.

Sometimes I see commands that end in an "e", like Zeige. Is that also correct?

Yes, Zeige is also grammatically correct. In the past, adding an "e" to the end of du commands was standard. Today, dropping the "e" is much more common in spoken and everyday German, so Zeig sounds more natural here.

Would a doctor actually say this, or would they tell me to "stick out" my tongue?

A doctor might say this, especially to a child (since it uses the informal du). However, the specific verb for "sticking out" a body part is rausstrecken, so you will also often hear Streck die Zunge raus! (Stick your tongue out!).