What will happen to the Indian plate after it slides under the Eurasian Plate?

It is a continent-continent collision following subduction, no subduction is taking place at the moment. The ocean floor that was attached to it has vanished into the mantle or got wedged into the thickened crust. The collision builds up immense stress, resulting in deformation, faulting, folding, crustal shortening and uplift of the Tibetan plateau.

https://www.files.ethz.ch/structuralgeology/jpb/files/English/Asiaeng.pdf

Will it become part of the Eurasian Plate?

That's a question of definition of plate boundaries, fault lines, sutures ...

Will it slide back out of Siberia?

Of course not. How could rocks even be preserved in such a process? In the process of shortening the crust is near completely consumed.

Will it get pushed into the mantle?

No. Continental crust doesn't subduct. Both sides have approximately equal density and the mantle is much denser, so they just push and shove each other. Maybe parts get pushed to greater depths (~100km) in the dynamic process and exhumed again. Shortening will continue, uplift, faulting, folding and erosion in equilibrium, and when and where uplift stops erosion will take over.

Afterwards there may probably be something like the Appalachians, an older result of a continent-continent collision and a very thick continental crust. Pop-science link depicting the principle: https://www.nps.gov/subjects/geology/plate-tectonics-collisional-mountain-ranges.htm