Mario Luigi Superstar Saga is a game like this with lots of gameplay involving gravity. It's basically a 3D gamestate with 2D graphics.
The Game Boy (Color) Zelda games, (and Minish Cap) allow jumping. For the Game Boy Zeldas, the height of the jump basically doesn't seem to matter. (There is some cuteness with raised switch blocks.) If you're grounded, you'll fall in pits and collide with enemies, if you're airborne you won't. All solid collision detection works the same.
I've programmed both types, I just used variables called Z, and Zspeed in addition to my X, Y, Xspeed and Yspeed. I do with Z and Zspeed what I would normally do to Y/Yspeed in a side scroller. To render the sprite, you just subtract Z from Y to get the right vertical position. Shadow rendering in the Game Boy way is just draw the shadow at X, Y.
If grounded and the jump button is pressed, set Zspeed to jump strength, set player to airborne.
While airborne, add gravity to Zspeed, add Zspeed to Z.
If Z is less than 0, set player to grounded, set Zspeed to 0, set Z to 0.
That'll get you to the Game Boy Zelda way. As eishiya mentioned, collision detection and representation of the collision map gets more complicated to do something like Mario Luigi Superstar Saga.