There are downsides, but not that most users will ever encounter.
Suspension lifts give you extra flex as well as extra height, body lifts just give you wheel clearance. If you are doing some serious off roading, obviously more flex is better. The blocks are also under quite a bit of stress if you are taking hard turns, hitting dunes hard, etc. Taller blocks = bigger levers = more force applied to the bolts, blocks, and attachment points on the frame and body.
Cosmetically, they also create a gap between the body and the bumpers (or just the rear bumper, depending on the truck and the kit) that looks like poo if not property addressed.
Generally you don't need extra parts if you stay with a smaller kit. Taller blocks may require longer brake lines and adjustments to the steering line though.
If you don't plan on doing anything real serious offroad with your truck (like dune jumping) and keep the body lift to a reasonable amount you'll probably never really notice the downsides, though, and you'll enjoy the upside of leaving more $$$ in your pocket for other mods.