I was looking into reasons why iTunes would keep prompting you to authorize items and still not play them, and found a great little article. It is named iTunes repeatedly prompts to authorize computer to play iTunes Store purchases found here: http://support.apple.com/kb/TS1389.
This is the overview of things to try but each one has its own set of instructions.
All the very best,
I have the same issue and followed all of these steps to the letter but still no dice.
It started this week and I have been using the same apple ID since iTunes began.
It only affects a few songs, and I have noticed that when I go to the iTunes store and am logged in with my apple ID these songs are greyed out and says that I have already purchased this item but still won't let me play it.
I have been scouring the forums and it seems there are more than a few of us having the same issue.
Also of interest is that these songs still play on all of my iPods.
Please fix this!