I would say this is extremely common in comedy, and much more difficult in drama. And even in comedy, a likable character can win sympathy very very easily even if he's totally unsympathetic - in fact, the moment he does anything that isn't actively unsympathetic, we'll probably find him sympathetic.
The Magnificent Bastard
As extreme examples, consider Barney Stinson from How I Met Your Mother, or Quark from Star Trek: Deep Space 9. They're very likable characters - we love them for their humor, their audacity, their ingeniousness. But at the beginning of both shows, we're not meant to sympathize with them, and we don't. They engage our interest, not our emotions. The way they're written, we're not saying "Oh, poor Barney!" or "Wow, Quark must be so nervous now!".
But once that's set up, sliding the characters into sympathetic territory is very easy. Season 1 of How I Met... has an episode where we discover Barney's past, which is hugely sympathetic. Later on in the series, his loyalty to his friends, his eventual love interest, his yearning for a father are all trickled in - and the mere mention of these, no matter how despicable Barney continues to act, and no matter how warped his attempts to address these issues are, are enough to convert him into a largely sympathetic character. That's because we like him so much - we're willing to forgive all his unsympathetic-ness as soon as we get the chance. Similarly, Quark has a long arc of growth through ST:DS9, and pretty much as soon as he starts having a good, soft, sympathetic side, and as soon as the show starts portraying him in a sympathetic fashion, he immediately becomes a sympathetic character despite his numerous and continuous misdeeds.
House, IMHO, almost comes under this category. But not quite. Because House, almost always, has motives and desires that we have strong emotional reactions to. When he's being an ass, it's often to someone we feels kinda deserves it. When he's consumed by a case, sympathy with his obsession for truth and discovery is key to the show. And when he plays mind games with the people he cares about most, he's often cruel and baffling, but he also generally has a motive that the viewer is meant to sympathize with. I think the show is based very strongly indeed on viewer sympathy with House - and the tension between that sympathy and where House's life choices actually lead.
Nonhumor examples might include Littlefinger and Varys from Song of Ice and Fire.
The Likable Schlub
A less extreme example is sitcoms. Sitcoms are full of characters we like, who do stupid things. It gets the plot moving. We're not supposed to sympathize with these characters - their problems are exaggerated; deliberately goofy and blown out of proportion. We're not sympathetic to Ross when he forgets his kid on a bus, or to Seinfeld when he preys on a fresh divorcee, or to Marshall when he insists he's going to prove the existence of UFOs. We like the characters, but at the same time, in these situations we're laughing at them, not with them.
Toeing the Line
Making this work with less-exaggerated, non-comic characters seems to me very, very difficult. If a character is likable, then all he needs to do is act reasonably OK, or with some clear, comprehensible rationale, and he's sympathetic. If he's likable, even a previously-unsympathetic character turns immediately sympathetic the moment such a rationale is presented (or take on a new goal which is reasonable). The Magnificent Bastard can, if you choose, avoid this by simply never doing anything sympathetic. But I can't think of other strategies or examples for doing this differently.
We are willing to sympathize with somebody we don't like, but we're strongly inclined to sympathize with someone we do. Overcoming that, it seems to me, requires constant avoidance of sympathy, and (depending on the extent and the scope at which you want to avoid the character being sympathetic) that severely limits the type of character you can construct.
However, this does suggest a strategy for having such a character for a short stint - take a likable character, and have him serve as an antagonist or an obstacle for some portion of the story. Portray him as being difficult and perhaps villainous - don't provoke reader sympathy for him in any way; don't emphasize his motives or his emotions. For this portion, the readers will not sympathize with him; he is in their way, he is unsympathetic (right now). But he'll still be likable (and, in fact, you can even portray his "good features," things readers like about him).