In almost all cases, no. Things like the Zune music player keep playing in the background, so they use battery, but most "normal" apps don't run unless they are visible on screen.
Some apps have the ability to get short bursts of execution time every half hour, but only if they get your permission. (Start Settings, swipe sideways to Applications, tap Background tasks.)
These background tasks are often used for updating live tiles that need up-to-date handset-aware information. (Tile updates that come from servers don't have current information about the phone, only what was provided when the tile "subscription" was established. The ones running as background tasks have access to current info, like GPS (for updating weather tiles), etc.
One of my apps has a backgroun task that gets info from a university server to update 4 tiles to let students know when there are course announcements from profs, when grades are posed, how much money is left on their meal plan cards, etc. You can do a lot in 15 seconds, but sometimes even these updates can't complete in time.