Please can somebody inform me what my courier is supposed to do to get rid of Motor Runner's goat's head helmet out of his inventory..?

Its like this with most quest items, it is impossible to get them out. I do however think that you can deliver it to some NCR guy at Mccarran but might be that I am wrong.

Yes, turn it in to Colonel Hsu at McCarren. Dunno if you had to have the quest first, or not...