When high frequency current goes through a superconductor, does it still radiate off like it would with copper?
Yes. Regardless of the resistance in a conductor, when a time varying current flows, it gives rise to a time varying magnetic field outside the conductor. When conditions are right, that time-varying magnetic field will give rise to a transverse electromagnetic wave (TEM).

In other words, an antenna constructed from super-conducting elements would still function as an antenna. The fact that the series resistance is zero does not prevent the antenna from functioning.