Is Ian (Damon) and Nina ( Elena) dating in real life?

Yes, they are.
I saw this on a website:

Vampire Diaries co-stars Ian Somerhalder and Nina Dobrev Dating?

Vampire Diaries co-stars Ian Somerhalder and Nina Dobrev have sparked romance rumors. The duo looked mighty cozy on a promotional trip in London.

Hollywoodlife called in an expert to look the chemistry.

“The chemistry is there, but they are doing a good job trying to hide it,” Leading body language expert Dr. Lillian Glass told them after looking at photos of Nina, 21, and Ian, 31, courtside at the Lakers game May 27 and shopping together in Venice, Calif. May 28. “She’s very into him. There’s no question about it. She’s always looking at him, or leaning toward him. They can hide all they want, but the body doesn’t lie. The truth comes out.”

“His hands are in his pocket [while they are shopping], which means he’s hiding, and his thumb is down, which means he doesn’t want to reveal anything,” she said, explaining, “He’s tight-lipped — it means he’s holding back. He is trying to deny the relationship.”

They are dating as they were walking through Paris holding hands in June 2011 and were pictured kissing at the airport in late 2010. Even though when they started doing TVD, Nina said 'if i wasn't dating Ian I'd be dating Paul' trying to say the media would think they were dating. If they weren't dating, they would probably have said something via Twitter by now tbh.

Um no, I dont think so. And plus, he already has a girlfriend.

No they're not dating he has a girlfriend called Megan and they've been together since 2006.

Yes, they are dating they appeared together as couple last month and Ian touched her both. Ian broke up with Megan a year ago, and then everybody said that he was dating with Ashley Green.

yes they are.

Yes, they are dating.

Yes.

Yes they are, i believe.

don't think it matter