I am stumped on this one, according to the link John posted above it seems that you would first need permission from the copyright owners but with it being short clips of music it seems a bit excessive and a waste of time having to go and do that. I have read that pubs need licenses these days to play music in some countries but I dont know if this would apply to yourself, I think it would really only become a problem if you started making money off the videos if you become a YouTube partner.
It is a shame things like this even exist because it isnt like your being malicious or would be really hammering someone else's profits. I would be put off doing them myself but it really is only a decision you can make yourself and knowing that there could be issues in the future, you could try humming them or do something like what they used to do on Shooting Stars years back where they would jokingly do song impressions, I know that wouldn't send any issues your way and it might make them more fun to watch
