According to the character's actor, Jim Parsons the answer is a confident "no", however he does happily admit (and has extensively spoken about) Sheldon's 'asperger's-like' characteristics.
JP: When I was first asked about it, I literally hadn’t… Well, I’d heard of the disease. Do they call it a disease? I don’t want to be…
AVC: A disorder.
JP: Disorder, thank you. How ridiculous now, looking back, that I said that. I’d heard of the disorder but I didn’t know what it was at
all. And when I asked the writers if Sheldon had Asperger’s, they
said, “No, he does not. That’s not what we’re doing.” - AVC Interview
Q. As the show’s gotten bigger, you’ve been adopted by more than one
community. Rather than say “Hey, do you want to be a spokesman for,
you know, Pepsi and we’ll pay you X?” people say, “You are now the
spokesman for Asperger’s.” How do you deal with that?
JP: Asperger’s came up as a question within the first few episodes. I got
asked about it by a reporter, and I had heard of it, but I didn’t know
what it was, specifically. So I asked the writers—I said, “They’re
asking me if Sheldon has Asperger’s,” and they were like, “No.” And I
said, “OK.” And I went back and I said, “No.” And then I read some
about it and I went, OK, well, if the writers say he doesn’t, then he
doesn’t, but he certainly shares some qualities with those who do. I
like the way it’s handled. - Adweek Interview
Similarly, the show's co-creators Chuck Lorre and Bill Prady head
indicated that from a writing perspective, the decision was taken not
to have him positively indicated as having a specific condition
because it would make it easier for them to change his character if they needed
"We chose not to diagnose Sheldon," says "Big Bang" co-creator Chuck
"Big Bang" co-creator, Bill Prady, who based Sheldon on computer
programmers he used to work with, has said they were afraid that if
they labeled Sheldon an Aspie, they would have too much responsibility
to depict the condition accurately within a sitcom. - NJ.com
Prady also stated in an interview with Slate that Sheldon's quirks are uniquely "Sheldony" without necessarily being part of a wider condition.
"I just think of his actions as 'Sheldony.' Some things feel instinctively correct for his character," says Prady, who recalls one software colleague who couldn't go anywhere alone that he hadn't been to before. "He'd say, 'I can't go to 47th Street Photo by myself.' And it was maybe three blocks away. It was never questioned. Quirks were never challenged—they were simply accepted as a quality of the person."
"Are these things Asperger's?" he asks. "I don't know."