Sound like "image persistence". But if it only takes a few minutes (as opposed to weeks or months) to leave the image then that monitor has seen it's best days behind it. There is no permanent fix.
Here's apple's document on persistence:
But as I said, this effect usualy results in keeping something in a fixed position for a fairly long period of time. Not minutes.
I have the exact same problem (might even be the very same 23" ACD model) and have had image persistence issues for a few months now. My browser's address bar is clearly visible against any flat color such as a wallpaper. I tried for months to clear it by using the method described on the Apple support article (full-screened white image) every night (approx. 8 hours) for a week and it did nothing. I also tried a screensaver called LCDScrub that has various different patterns designed to clear image persistence, but that didn't help either.
A few days ago though, I noticed that the persistence "shadows" started appearing really quickly; it only took a few minutes after I added a new bookmark to my bookmark bar for the logo to appear at that spot onscreen and despite trying the remedy again, it hasn't vanished.
What is the policy for this? I've heard things ranging from manufacturing errors on certain ACD models which highly increase the chance of IP happening, to "tough luck, it just happens" basically. I know the LCD ACD's are older pieces of hardware but since this issue isn't exactly caused the by the user or any kind of misuse of the display, are there any chances to get a replacement or anything like that?