If the other end of that wire is still connected to something you can't just shove it capped or taped up into the wall. Your options are to find and disconnect the other end (in which case you can just cut off or otherwise dispose of the wire as you wish) or to put a box up there and terminate those wires in the box. You can put a blank plate over it and paint or otherwise camouflage it.

Thanks, Ron. I knew there had to be something more to it, but my husband was sure he could just push them up into the ceiling and be done with it. I told him before I posted that I didn’t want to push them up in there and later find out we created a fire hazard. We will find the other end before proceeding. Thanks again!