I haven't put together a coherent feeling about American vs British irony, so I'll back away from that for now.
One observation I would like to propose about American TV, however, is the best examples of funny TV we've seen listed here are ironic, not for what is said in any given episode, but more for the structure and format of the shows.
The Simpsons is a cartoon, fer chrissake, in the style of The Flintstones, but look at what they're doing. Shoddy nuclear plant in town, dad's a drunken lay-about, etc. Also look at South Park, which, if anyone hasn't seen it, is very much in the general style of a Charlie Brown cartoon. In the South Park world, however, the adults are crack-whores, homicidal weirdos and closet perverts. The kids are not sweet and slightly philosphical. They're mean, nasty and cruel, and yet they manage to come out with some thoughtful observations on the world.
During the early 1980's when the NBC network ran a set of family-based sitcoms (Cosby and Family Ties), Fox responded with Married with Children, the anti-family show.
For me, the fact these shows ran at all is the irony.
Rich