I am spatchcocking a 16 pound turkey. I can either try to fit it in the oven or grill it on a charcoal Weber. Which would work best so it's not dry?

You can treat your grill just like a large outdoor oven. For large pieces of meat like this you generally want to go with an indirect heating setup. with the lid on. (http://www.weber.com/weber-nation/grill-skills/mastering-basics/charcoal/fire-configurations)

You'll be aiming to pull the turkey when the breast meat reaches ~165 and the thigh meat ~180. If you position the turkey so that the legs face the coals, you should be able to make it so both parts of the turkey hit the desired temps at the same time.