The first problem I thought of is that as the speed goes up, the centripetal force causes the pressure to increase. At some point there will be a transition from liquid to solid. Eg, for ⁴He, and if I read the phase diagram correctly, that's at 25 bar.
I searched for "rotating superfluid helium" and found http://physicsworld.com/cws/article/news/2018/feb/07/unexpec... , which is a summary of https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.01... . It says that there is friction, perhaps caused by "quasiparticles that become trapped within the cores of vortexes. As the vortexes accelerate, the quasiparticles gain energy, which they can then dissipate to their surroundings in the form of friction."