Why does it hurt my throat to stick out my tongue?

Muscles: The attachments of the tongue muscle [and the tongue is essentially one large muscle covered with sensory tissue] is in the back of the throat and in what is called the "oro-pharynx. When you stick your tongue out, you are stretching those attachments. As far as it hurting, if the throat is sore from other reasons [trauma or infection] when the tongue is stuck out it will irritate it.