The term for a song that people quickly become tired of on the radio?

Solution 1:

I think your best bet is to say that the song has been overplayed on the radio (or in other words, an overplayed song). While I don't consider Urban Dictionary to be a reputable source when it comes to definitions, the one they have posted certainly fits the use of the term:

used to describe a hit song that is played on the radio (in stores, on the internet, on iTunes, at parties, etc.) numerous times to the point of annoyance

Solution 2:

I think the term used was probably burnout. This is one of the things measured in radio "callout" surveys of listeners (all bolding added):

Another type of data often obtained during callout is a fatigue or burnout measurement. Listeners are asked, "are you tied of hearing this song in the radio?" (The Concise Encyclopedia of American Radio, edited by Christopher H. Sterling, Cary O'Dell, 2010.)

If the host was talking more about why some songs lose their zest more quickly than others, he might have been talking about something like this:

Add all the scores together [rhythm, Melody, etc.] and you get a complexity score. Divide the score by the number of times the song will be exposed to it's target audience and you get the sing's burnout index for that audience on your station. The higher the score, the longer the song should take to burn out. (David MacFarland, Future Radio Programming Strategies: Cultivating Listenership in the Digital Age, 2013.)

For the question specifically in your title, a song that people get tired of quickly could be said to have a high/low/fast (depending on usage) burnout score (or rate).