We've found easy methods for hiding the icons of running apps in the Dock, but you typically have to get your hands dirty with editing files inside the app package itself. For users who prefer a more simple drag and drop operation, there's always Dock Dodger. From the same FoggyNoggin Software that brought us Desktopple Pro, Dock Dodger handles the dirty work of ensuring your app never shows its icon on your Dock; simply quit the app if it's running, drop it on Dock Dodger and restart it for the ultimate in 'what am I going to do with all this open space in my Dock?' experiences. While Dock Dodger performs the same editing of app files that you can do yourself, we'll still echo their recommendation of duplicating the original app first so you have a backup just in case.
Dock Dodger is available as donationware from FoggyNoggin Software.