I'm afraid the only way to have this fixed is to have someone look inside and check things out. It sounds like the contact inside is registering twice or there is a short somewhere in the wires leading to the disc tray.
Your best bet would be to contact Sony's tech support and see if they would be able to fix it. This would require sending it in to them and you would have to pay for the repairs- not sure how much that would be but they can tell you before you make the decision to do so. Depending on how much it is it may be cheaper in the long run to pick up a new slimline PS2 and get yourself the warranty on it.