Leg or brown meat on turkey I believe makes better burgers, purely because the meat is juicier. I don't want to get to tecci, but its all to do with the leg muscles working harder, and applies to all white leg meat, so pork, chicken and even game.
Breast meat is fine also, and is extremely good for you, but needs a little more attention when preparing and cooking. Skinned breast of turkey can be less that 2% fat, so when it cooks, it can cook slightly drier, leaving you with a crumbly texture. But I have discovered a great trick. If you add a little mayonnaise to the raw mix it really helps to keep the burger beautifully juicy and tasty. You only need a very small amount, as you can see below.