Bought my system a week after launch, set up WiiConnect24, noticed some glitchy pixels after the New Year while playing games like Zelda: Twilight Princess where layers of fog/dust appeared in the foreground. Never saw the problem with simpler games, but it did turn up in Godfather: Blackhand Edition. Never encountered any actual gameplay glitches, it was strictly a display issue.
Sent it in for repair during the May drought, since then no problems at all with WiiConnect24 always on. I don't think the problem has anything to do with WiiConnect24, it seems related to the level of graphical intensity and some Wii GPUs had faulty display buffers.