I fixed this error. Reason - badly placed laser.

Alright, I've just replaced the laser on the PS3, because it couldn't play any games (was already out of warranty). Now whenever I try to launch LBP 3, I have for about 4 sec black screen and then get kicked to XMB with error:

'An error occurred during the start operation.
(80010514)'

I see that laser is not even moved up or down to load the disc, which is really odd and at the same time I'm mad at PS3 because it's too lazy to load the game

Please help, I've restored System Files from Recovery Menu but it's still not launching

Here's link to my video of it: https://youtube.com/watch?v=KdQOAsj5ToY

It's most likely your PS3. Have you tried any other games?

LBP 2 plays fine, but I need to move the laser by myself because at some point it can't move itself, can't launch LBP1 because it's impossible to me to move the laser that needs to be probably at the top of the disc. Haven't tried any other games that I have, but guess I have to.

Only disc that is read by the PS3 perfectly is PS Move Starter Disc and it doesn't need extra help.

Did a little research on this and people say it's the lens is failing on you. Honestly you're better of replacing the whole blu-ray drive in the PS3. Depending on your model, it's not that hard to fix, though a little annoying. I had to replace it for my friend in his Slim PS3. Not entirely sure about the fat PS3 as you need a special screw drive to pop the top.

So easiest solution is to buy a new PS3 while the harder solution is popping open your PS3 and replacing the blu-ray drive.

I've been searching the whole disc drive for my PS3, but I can find seperate parts only ;(. I wanted to buy new PS3 though, but the parents didn't want to D:

Now I found that LBP1 works as well as LBP2. Still LBP3 won't launch :/ I forgot also to tell I have PS3 Slim, not Fat. For some reason in my friend's Super Slim LBP 3 works fine, but not in my PS3 Slim

Here's link to my video of it: https://youtube.com/watch?v=KdQOAsj5ToY

That's weird, I was going to suggest trying another copy of LBP3 but if it works in your friends PS3, then that can't be the issue. Another issue could be something with the hard drive as it's been known to glitch out games. You can try booting into safe mode and restoring the file system on the PS3.
JUST BE WARNED THAT THERE IS A POSSIBILITY IT MIGHT DELETE SAVE DATA. SO BACKUP BEFORE YOU TRY IT!

Edit: Before trying all that, delete the game data for LBP3 and try reinstalling it first.

Tried everything you suggested. Still doesn't work.

It sounds like a physical fault with the Blu-ray drive. You could replace it, they go on Ebay for about $50.

As a last ditch effort you could try cleaning the laser lens, making sure the data ribbon connecting the drive is secure and giving it a general once-over.

Hold on. I was waving my PS around and it launched! At first try failed but at second LBP 3 was installing! but accidentally unplugged laser ribbon and have to start again

I discovered that laser was too far away from the disc so it couldn't always reach the data. I've put some tiny things to raise laser a bit. Not only error code dissapeared. This physical issue I've mentioned has also gone! And from now on, my PS3 even started to play game that wasn't earlier recognized before - Uncharted 3! Right now I'm very excited about PS3 being as good as new.. and of course because I can finally play LBP 3 at my home!

..aaand yup GTA V leaves still black screen. Guess I have to install the game again

You haven't helped a lot but thanks that at least you tried!