I should probably update this to say that on the 30th August I received a refurbished Xbox 360 from Microsoft that not only works (no problems thus far, and I've had many a 5-hour gaming session on it), but it also considerably quieter than my last one. So it would seem that contrary to what I had read elsewhere, Microsoft are still sending refurbished units out instead of fixing and returning people's original console. For me, this was a good thing, but for others I can imagine it wouldn't be.
Well, here we go again.
Let's start from the top (this time). A couple of months ago (late May/early June) my Xbox 360 started displaying a bright red/pink tint over games for varying periods of time. The exact times it occurred depended on the game, but it was more frequent in Oblivion than in Crackdown, and in FlatOut it only occurred for a few minutes, approximately 10 minutes after first starting the game, after which point it was fine. In some games it was even very clear that the game's HUD was unaffected by the tint, while the rest of the game was (though in some games, both were affected with graphical glitches in some ways - HUDs, including the Dashboard, gained faint vertical lines).
Witness some examples below: