Saying "la tête me fait mal" is more natural to a French speaker than "ma tête me fait mal". You'll use the second version only if there is a potential for confusion about which head you're talking.

I understand but when I answered "la tête me fait mal", it was marked wrong. 

I suspect this part of the lesson is relevant here:

You use a possessive adjective when the body part is the subject of the verb. "Ma tête me fait mal"  is actually given as an example in the lesson.

OK but why is "me" necessary if you use the possessive pronoun?

Because the pattern is: qqc fait mal à qqn. 

When "tete" is an object, it is preceded by the french definite article and a preposition. This is a generality when you are talking about parts of the body.

J'ai mal à la tête = I have a headache.

Je suis blessé à la main = My hand is hurt.

Chris 

Surely "qqc" can mean both "la tête" and "ma tête" in your construction.

This was to explain why to use "me", as per your question. The French "me" corresponds to the "à qqn" part.