Learn German: Mein Knie tut weh.

They come from the separable verb wehtun (to hurt). In a standard main clause, the conjugated part of the verb (tut) takes the second position, and the prefix (weh) gets pushed to the very end of the sentence.

Is Knie masculine, feminine, or neuter?

It is neuter (das Knie). That is why the possessive word is mein without an extra ending, since it is the subject of the sentence in the nominative case.

How would I say this in the plural if both knees hurt?

You would say Meine Knie tun weh. The plural of das Knie is die Knie (spelled the same, but pronounced with two syllables). You also must change the verb to the plural form tun and the possessive word to meine.

Does tun always mean to hurt?

No, the verb tun by itself just means to do or to put. It only means to hurt when it is combined with the word weh (which means pain or ache) to form the separable verb wehtun.