My history:

1. Fresh 12.04 installation
2. I was able to mount and play a commercial DVD
3. I inserted a DVD that was apparently scratched. It didn't work
4. Now DVDs won't even mount

Drive: Optiarc DDU1681S
3.2.0-26-generic x86_64

With the DVD drive open. trying to 'eject' it closed:
eject: unable to eject, last error: Inappropriate ioctl for device

Manually close the drive. In dmesg:
Buffer I/O error on device sr0, logical block 0

The light on the drive blinks steadily, and the hardware eject button is ineffective. 'sudo eject' works.

I also used regionset to set the region to 1 (my region). Putting in a CD works normally.

Thanks for any advice!

Did you mount it manually or auto mount it with udev or hal?
Is the DVD still mounted, even though you took the disk out?
Will show you what's mounted. /dev/sr0
If it's still mounted try a force umount on the mounting point. That's where ever you mounted it to.
You might put it right by reboot

(No disc in drive)

$ eject
eject: unable to eject, last error: Inappropriate ioctl for device
$ sudo eject
eject: unable to eject, last error: Inappropriate ioctl for device

(Use physical button to eject and put in a DVD)

$ mount | grep sr
$ mount /dev/sr0 /dev/dvd
mount: only root can do that
$ sudo mount /dev/sr0 /dev/dvd
mount: /dev/sr0 already mounted or /dev/dvd busy
$ mkdir ~/dvd
$ sudo mount /dev/sr0 ~/dvd
mount: no medium found on /dev/sr0

Looks like it wasn't unmounted last time. Umount it manually.
Don't open the DVD drive until the disk is unmounted. You may have to force umount it. The drive may be hung do to ejecting the disk without umounting first.

http://linux.die.net/man/1/eject
http://linux.die.net/man/8/umount

I you are auto mounting you'll have to look at what your distro is doing, you may ask in your distos sub forum. If you are mounting manually, umount the disk, then open the drive. If you have a junk DVD, it may need to be force umounted, or the drive may get in a confused state. I'm not sure that eject will umount the file system before it opens the door.
Also a reboot may cure the whole problem.

I may be wrong but isn't trying to mount /dev/sr0 on /dev/dvd always going to fail? To manually mount a DVD I would aleays do something like:That's assuming that your DVD is /dev/sr0 and not /dev/dvd and that you have an appropriate folder in /media/.

Thanks for the advice! How can I look at what my distro (Stock Ubuntu) is doing? Here's a log from just now. let me know what else may help:

Restarted my system. Nothing in the drive.
Drive opens. I put a CD in the drive and press the physical button to close. I see the CD show up on my desktop and mounted at /media/cd-name.
Drive opens. I replace it with a commercial DVD. The light on the drive blinks steadily. Nothing appears on the desktop or under /media
The drive opens.

Thanks for any help - let me know what might help. It definitely seems to be DVD-specific.

Are you 100% sure you installed the non-free packages to read CSS DVDs?
I ask because I've done this before and installed Ubuntu and missed the specific codecs.

I'm not sure

Then I also ran the install-css.sh script

What else may do the trick?

That ought to do it. It's worth double-checking is all.