Did you buy the console used? If it was doing that when you bought it, than it may have a "Lost DVD Key." Xbox 360 DVD drives are "Married" to the motherboard of the console that they came with. If you simply replace a faulty drive with a working one, the console will not read game discs, only DVDs and CDs. You need to either repair the original drive, or swap it's controller board with a drive of the same model. (There are a few different drive variants, so you need a drive of the same model) If repairing the drive / putting it's controler board into an identicle drive doesn't solve your problem, there is really no possibility of regaining game functionality (unless the console is on a very old version of the dashboard, but I'll go into that more, if it's nessicary)
Unfortunately, you can't just swap dvd drives in an xbox 360 and have them work. You have to replace with the same type of drive and retrieve the dvd key from the original drive and spoof it to the new drive. It sounds harder than it is, but it does require hooking your xbox drive to a desktop computer with a via sata pci card.
The problem is not only that the drive and motherboard are cryptographical
There seem to be a lot of checks being made between drive and motherboard, that will cause your console to not being able to update if you have a replaced drive.
The last update in spring 2011 was specifically made to detect modified drives, and in turn a lot of consoles with swapped out drives would not be able to finish the update, as the altered firmware on the replaced drive was not properly flashable by the update.
So essentially, if you want to continue to play the latest games, that require an updated firmware, do NOT attempt a self repair.
I had the same problem, where my DVD drive wouldn't play anything at all. I swapped drives and movies would play but not games. So I swapped the Printed Circuit board from the old drive to the new and it worked just as new and no need to flash. Just make sure your replacement DVD Drive is the same model as your old. There's tons of you tutorials on how to swap PCB on youtube. Keep in mind that some models involve soldering but thats easy as well. NOTE: My Xbox in the old FAT one NOT the slim. No sure if this would work on a slim.