Do I need rim joist for this tree house platform? What will happen if I leave them off?
Do you have to?

No.

What will happen if you don't?

The unsupported ends of your joists will:

sag
twist

The decking on the left of the single tree (from the angle of the picture you provided) will become uneven, making tripping hazards for the occupants up top. You'll also end up needing to replace those deck boards more frequently, since not only will they want to do some warping and cracking of their own (it's wood, that's what it does), but they'll be forced into even more warping by the joists moving underneath them.
An additional consideration - without the rim joist, what are you going to attach the railing to? Having the nice straight, even surface of the rim joist gives a good mating surface for screwing the railing posts to (with some serious sized structural screws, not decking screws (they're not designed for that kind of load), and especially not drywall screws). Without it, you're going to have to fabricate some pretty unique and beefy brackets to hold the railings to the ends of the joists. You'll probably have to cut through that last deck board to attach them. Then, you'll get to watch the whole railing get ripped apart as the joists twist with age.
But, you don't have to.
Maybe OP had thought about this, but it just occurred to me: I'd put a rim joist at the far end, too, even though each joist is directly supported by the beam and attached to them with the hurricane clips (kudos for that!). Those metal clips will help prevent the bottoms from moving when the joists want to twist (if it twists away from the metal, a joist can pull itself off the nail, leaving the metal behind), but will leave the tops free to move. A rim joist on the far end will help prevent warping all the way along the ends of the joists.