If you say "the videotape clicked on" it suggests some kind of autonomy or separation, by making the videotape seem like an agent of action.
This separation could be extreme to the point that it is actually suggesting that "the videotape clicked on (by itself)" -- perhaps because of a timer. That would give it some of the ominous overtones mentioned as being in a mystery or horror novel, as if something spooky has just happened. But you could also say "He pressed the remote button and the videotape clicked on."; which puts a distance between the remote and the videotape--as if "clicking on" was a choice the tape was making.
"Clicking" here can suggest a noise is made by the machine itself, and certainly mechanical tapes had a THUNK as the playhead engaged. Kids today will not be familiar with that--though perhaps they would know the "whirr" of a DVD spinning up. Next decade, maybe that sound will not be familiar either. :-)
But as @magistermurphy says, the "click" could merely refer to the moment of discontinuity between being off and then on. In fact--even if something had no sound but went from a dark screen to a moving image you could be understood in saying it had "clicked on".
A phrase like "could you click on the telly for me?" sounds a bit like something an old British person might say, along with the humorous "I'll knock you up in the morning.". It's colorful if you want to play with language, but not the best way to phrase things if you're trying to communicate clearly.