Got bad news for ya - I've seen this happens on two of our Mac Minis in our creative department, also iMacs, notebooks etc.
Your drive is dead, kaput. We have tried everything, reset PRAM, SMC reset, taping the front, etc. All solutions doesn't work. Apple tech later affirmed that the drive is hardware wonky.
I suggest you call Apple ASAP if your warranty is still on. If not, you might want to swap it out for a third-party replacement you can find online, like this one.
Slot-in drives are usually much more susceptible to break-downs than tray drives. Sometimes I resent Apple for choosing form over functionality for certain hardware selection.
If I were you, I'd replace the drive on the Mac Mini with another hard disk, and use a USB based DVD burner. If you buy another slot-in drive, I can't guarantee it will last you long too.
EDIT : QUestion updated to reflect it is only DVD+R writable that cannot be read. Above answer might not be valid anymore. New answer below.
Are you sure you have a SuperDrive capable of burning DvDs? Or could it actually be a Combo drive, capable only of burning CDs?
To find out, go to the Apple menu ->About this Mac ->More Info and up pops the System Profiler. In the left column should be a heading called "Disc Burning". Choose that and it should detail info on whether can you burn or read DVD.
Some slot-in drives are also very susceptible to low quality media - did you try with another brand of DVD+R?
Finally, there's Apple troubleshooting.
- Check the disc's surface for scratches and dirt, since these things can prevent the disc from appearing on the desktop.
- Reset the Power Manager (PMU) or System Management Controller (SMC).
- Ensure your drive can read the specific type of disc being used. For example a "combo drive" can read a DVD-R and burn a CD-R however it cannot burn a DVD-R. The utility System Profiler may provide additional information regarding what media your computer supports.
- If the discs continue to not be recognized, please contact Apple, contact an Apple Authorized Service Provider, or make an appointment with an Apple Retail Store for additional troubleshooting, evaluation, or service.