RSSIf I show my character watching TV, do I have to get permission show whatever is on the screen?
Internet Filmmakers' FAQ
Unless you know for sure that the clip showing on the TV is in the public domain, you should get copyright clearance to use it. Otherwise you may find that the shot has to be removed from your final film before it can be shown publically, or worst, you get a letter from the copyright owner demanding a license fee plus damages for the unauthorised use of the their material.
In some cases, particularly with documentary films, it may be possible to argue that the clip on a television constitutes "fair use" (as defined in copyright law). However, to be successful in this defence the clip needs to be incidental to the shot. You still may end up in court having to defend your assertation of fair use, but if the clip isn't featured or favoured in your shot, you'll at least have a chance of convincing the judge that fair use applies.
But the upshot is, if you have any control over the scene whatsoever, turn off the television or replace the content with something which either a) you have the rights to, or b) is in the public domain. If that's not possible, you should seek to clear the clip, or don't use the shot.