According to Google and the wiki error code 80010510 translates to: Will not play games installed on the hard drive
From what I've read, the problem could mean the PS3 Blu-ray drive is on it's way out, or it could be a bug related to the specific OFW itself.
One thing you can try is pressing the "PS" button while playing from PSN to see if the game disc shows up then.
Another thing I'd try if you have a second PS3 around is to see if those non-working game discs work on the other console.
Finally, what happens if you insert a non-game disc (such as a DVD movie disc) into the PS3, will it read that or nah?