Does beer suffer from being warmed and then rechilled? I typically prefer my beer only a bit cold, so when I buy a 12-pack from a store's cooler I typically just leave it out. The excess I'll refrigerate at the end of the night, and sometimes repeat the process with the same beer on a different night.

Does this affect the quality of the beer in some chemical way? I almost exclusively drink IPA's and personally never notice a difference, but many of my friends have commented on the habit of mine to let beer warm.
TL DR; No.

Beer flavor changes over time (hops fade away, oxidation takes hold, etc.), and this process happens more quickly at warmer temperatures than colder ones. But there are no additional chemical reactions caused by temperature changes, so warming to room temperature and re-chilling multiple times is not going to have any added effects on the beer. Assuming you are drinking it within a few weeks, you won't notice the difference with bottled or canned beer.

I think this myth took hold from left-over kegs after parties: A half-empty keg that was dispensed by pumping air into it will start to oxidize much more quickly since oxygen is being added to it. When it warms the oxidation speeds up and it tastes stale within a day or two. Keeping it cold slows that down a bit, but even cold it won't last very long. A keg being dispensed using CO2 is a different story: that will last as long as bottled beer and can be warmed and rechilled without ill effects.

In all of this, I am referring to room temperature. Beer left in a car in the sun for hours will start to stale much more quickly, even if bottles are protected from the sunlight. You won't get the skunky flavor caused by the light, but it will taste stale.