According to Roaring Apps, Illustrator CS3 may not be compatible with Mountain Lion, while they report that CS4 is compatible. I've CS6 and am printing from Illustrator just fine. I also don't see any 3100 series here - http://support.apple.com/kb/HT3669#HP (but check it again for yourself).
If it is supported, you should be able to download Apple's HP drivers and install your printer anew - http://support.apple.com/kb/DL907.
If no joy from Apple, try contacting HP.
CS3, CS4 or both? Only when you try to print? Download a 30-day trial of Illustrator CS6 - click "Try".
If that works, you'll know that your version of Illustrator just isn't working - and if it doesn't work, you'll know that it's a problem with your printer driver.
Thank you your suggestion to dowloand the trial and try printing from Adobe Illustrator CS6 worked.
Now I have to purchase an expensive upgrade to Adobe Creative Suite.
The HP Photosmark 3100 Series printer is on the list of printers that should work though . . .
just found this on a german forum:
delete the following Files/Folders and it runs again:
worked also on mountain lion with CS4!