Everyone else around him is in suits, whereas (assumedly due to his condition), he is unable to be constricted by suit pants, so he opts for jeans and sneakers! (often runners)
I think it has very little to do with his condition and more to do with his general rebellious attitude. Tell House to treat a patient one way, he treats them another. Tell House to wear a lab coat, he ignores you. I'm sure the same is concerned when asked to dress professionally; he dresses how he wants to dress.
I also think House's wardrobe works for him. Many here agree on the topic that wearing suit jackets with jeans looks like you're trying to hard; I disagree. If it works for you, it works for you. 99.9% of people don't give a damn if you wear a suit jacket or a sport coat with jeans. Arguably, wearing anything beyond your regular parka or zippered hoodie or jacket is going to show you have some style and give more than 2 cents about your appearance.
And a side-note for House fans: SEASON PREMIERE TONIGHT ON FOX AT 8PM
It all depends on the fit of the jacket and the person wearing it. I just wore a really nice grey jacket from my three piece with a v-neck over a dress shirt and my lucchese cowboy boots. If they were ugly boots, the jacket didn't fit, the jeans were baggy, or I wasn't the right look (some people just don't look good in certain things) then it would be bad.
And untucking a dress shirt looks better than tucking it into anything just because you think it can't be untucked. Again, it goes back to the fit. Not some yes or no rule. I can't stand seeing someone tuck a dress shirt into a pair of jeans, regardless of how well the fit is. Unless you wear a sweater over it.
I think it looks a bit goofy with a longer jacket, like the first of the two pictures above. And the stripe on the jacket just makes it looks like it's missing the pants. The second jacket looks a little shorter and appears solid, and I think it looks good, (especially with the white jeans).
So, like other people said - more modern/less classic solid jacket and this looks fine.