As an Apple-certified technician, I can tell you they can pop your serial number into a web interface and pull up warranty info on your system (I could as well), receipt or no receipt.
I bought an iphone from apple store but I lost my receipt. Is there a way I can get a new copy of the receipt either by calling them or visiting them?? And what do I need to provide in order for them to retreive the receipt? e.g. IMEI or serial number..?