Which is a temporary session file in the temp dir. It is just one issue ;)

Or how would you go about uploading the missing file /home/***/tmp//sess_049b4b10fabe8f735fce3255f17bebfa, O_RDWR?

The tmp directory/partition is probably full.

The tmp directory being full might be something, but the server wouldn't be performing well/at all if that was the case, at least long term.

I'm still nearly positive that he doesn't have permission to write to that folder.

Are you on godaddy by any chance? You should request to move to one of their new cPanel Linux accounts.

Ohh dude, there wasn't tmp folder in my home. I create a new folder and set permission 755. Errors gone, but facebook platform is not working :S

That doesn't address the issues with facebook retiring their api that the vBulletin 4 stuff was built on. There is not currently a fix.

Sorry for my bad english.
Is there a problem with facebook-vbulletin connection?

Yes facebook changed the API needed by vBulletin to connect to it, so it no longer works.