Other people with Windows 7 have not had an issue with their systems recognizing the DVD, so it is most likely a problem/conflict with that particular optical drive or system.
Did you try booting into Safe Mode (hold the F5 key down as Windows starts, or hit F8 for a boot menu) and see if the DVD is recognized there?
In the Device Manager, you could try deleting the entry for your optical drive, reboot, and let Windows reinstall the driver for it.
Did you try updating the firmware for the optical drive, if there is one available? Also, check Dell's site for any BIOS or motherboard chipset driver updates.
Do you have access to an external optical drive you could try on your system?
Do you have any computer geek friends that might have a spare optical drive lying around you could borrow? DVD drives are relatively cheap, but I wouldn't necessarily want to buy a new one if I wasn't fairly sure it wasn't a software problem.
If you are comfortable changing hardware, you could probably swap optical drives in the Vista and 7 systems (check that they both use the same connectors on the back of the drive), and see if the problem moves with the drives (in that case you should be able to convince Dell that the drive is starting to fail, and get it replaced under warrantee). If your optical drive uses a SATA connector (small connector/cable, same as the hard drive) then you might not be able to swap it; with a PATA connector (2" wide) you could probably swap with either the XP or Vista systems (SATA optical drives used to be less common than the older PATA drives a couple years ago; new systems may not have a PATA connector on the motherboard).
