I've had the same problem with my 27" I-mac. I telephoned technical support who immediately gave me a service ref No. and told me that the service company would call to arrange a pick up. Approximately 2 minutes after putting the phone down to Apple I got a call from the service company who arranged a pick up for the next day, asking if I wanted am or pm. They were on the doorstep at 8am the next day with a packing box to to transport the i-mac. The company has stayed in touch, leaving me updates and appologising for the delay. It turns out they got a new screen, but weren't happy with the fit, so sent for another. They tell me that they will deliver it back to me this morning.
They described the issue as screen burn. Although I am worried that this has happened on a machine that is under a year old, I can't fault the after care that I received from Apple and the service company.
Just out of curiosity, do you not have an Apple store nearby? Or will Applecare always give you this option to pick up and deliver?
Keep us posted about this "screen burn". I've read many posts about the same problem on the 27" screens.