Are the 2s and 2p subshells degenerate in the He+ ion?
Yes, because this system is identical to the original if we simply rescale the charge (or, if you'd like, modify the units of our new system). On a more fundamental level, degeneracies are caused by symmetries of the system, and these are affected not by the magnitudes of various constants or parameters, but rather by the form and structure of the relevant interactions.

The degeneracy of the subshells in the hydrogen atom corresponds to the SO(4) symmetry of the hydrogen atom, which leads to the conservation of the quantum-mechanical form of the Laplace-Runge-Lenz vector. This symmetry depends on the inverse-square law nature of the Coulomb force and appears also in celestial mechanics, where the corresponding inverse-square law force is now gravity.

Changing the charge of your hydrogenic atom will not break this symmetry. On the other hand, fine structure corrections to the hydrogen atom will: if you start looking at relativistic and spin-orbit corrections, you change the form of the Hamiltonian, break the symmetry, and hence resolve the degeneracy.