Will two 2x10 beams carry a a 2500lb floor for a tree house?
There’s several issues with your plan: 1) support 2500#, 2) connect platform to beams, 3) differential loading on cantilever beams. 

1) Yes, 2 - 2x8’s spanning 4’ or 5’ can easily support 2500# depending on the species and grade of lumber. However, a 4’ or 5’ cantilever is excessive for either a 2x8 or 2x10 and will eventually fail depending on how much of the 2500# load is on the cantilever beams. 

2) The platform could exert a force of over 600 lbs. due to rotation on each H-1 connector by having the live load and dead load on the cantilever. The Simpson H-1 clip is only rated for a resistance of 400# to 500#, depending on the species and grade of the joists and beams. H-1 clips are not suitable for this installation. 

See: https://www.strongtie.com/resources/product-installers-guide/h1-installation

3) Structures like this will not have uniform loads distributed throughout the structure evenly. A more detailed design and structural analysis should be done by a structural engineer (not civil engineer) or architect if you decide to stay with the extremely large cantilever design. 

Btw, there are many videos about how to connect beams to a growing tree for a tree fort.