.
The M16 bolt carrier has a longer "bridge" that encloses the bottom of the bolt carrier. This longer bridge section interacts with the Auto Sear which is needed for fully automatic fire. The presence of the M16 bolt carrier should not affect the performance of an AR15 as long as the AR15 has standard parts. The presence of the M16 bolt carrier can, however, earn you a fellony should your rifle ever be inspected by a knoledgable law enforcement officer.
If by cycling the next round you mean that when you fire a round with your semi-automatic rifle, the spent bullet casing is ejected, but the next round does not make it into the chamber properly, that sounds like an unrelated problem. The presence of the "bridge" in the M16 bolt carrier shouldn't cause this.
I wouldn't worry about selling a part that doesn't function. Just having that part around your house while you own an AR15 could be just enough evidence for some BATF agent to charge you with the intent to convert a gun into a machine gun.
If I were you, I would exchange that M16 bolt carrier ASAP, throw it away, or mangle it beyond any practical use. Just buy a new bolt carrier.
www.bushmaster.com/shopping/uppers/8448505-s.asp