Kirby can eat items thrown at him, but can't get any powers from it. When I swallowed the Lightning Bolt, it activated and shrunk everyone.
Some projectiles you can eat, like Link's Bombs. Kirby eats the bomb, and then the bomb blows up and he puffs up for a second. A Bob-omb caused me 5 damage, but I'm not sure if Link's bombs do the same. They cause the same animation though, so I think it's safe to assume they do.
Fast or sharp projectiles, like arrows, lasers, and Pit's light arrows, can't be swallowed. Neither can Link's boomerang, but what's fun is to let the boomerang go past you, then start inhaling. The Gale Boomerang then comes back, and brings you right to Link, letting you swallow him :D It's a fun surprise move.
No Jester hat as well
![Frown :( :(](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
You can't swallow an incoming Waddle; it just hits you in the face. But once they land you can swallow them, but you get nothing for it.
Edit: Went and did a bit of testing to make sure I was right, post edited slightly.