Are there any beer styles that are served flat? I thought of this question when I've read the one about mulled beer. I thought heating the beer would make it lose carbonation, but then I've read somewhere that you have to carbonate the mulled beer as well.

So are there any beer styles that are served without carbonation or nitrogenation? I myself think that there are probably none, but I'm no beer expert (yet :D).
Historically beer was almost definitely still (flat) for thousands of years. Before the discovery/invention of force carbonation methods, all beer was carbonated naturally via bottle or cask conditioning. But people were brewing alcoholic beverages commonly referred to as beer in antiquity, and evidence from these cultures (ancient China, Neolithic culture, etc) suggests that they were doing this in big stone and earthenware pots and jugs, which may have had no lids or loose lids. Their vessels probably could not have withstood the pressure of carbonation even if they were using wax or cork to seal the vessels.

So if you were to brew a historical style and you wanted to be truly accurate, then you would not carbonate it. But the modern historical interpretations usually are carbonated (like Dogfish Head's ancient ales), because they are selling them and Westerners generally prefer them :)

There is one "modern style" (e.g. a style you'd find in the BJCP style guide), that can be served with no carbonation, namely straight (unblended) lambic. Several have little or low carbonation, (e.g. barleywine) mentioned in other answers. Sahti also has very little carbonation.

But people do still drink flat/still beer all over the world, for example there's some uncarbonated corn beers that are popular, like chicha in Latin America (Dogfish Head made a Chicha inspired beer) and umqombothi in South Africa. There are other names for similar beers in other regions. These are typically homebrewed.