Turbo fans do indeed run with super-sonic tip speeds. The RB211s typically had Mach 1.4 at the design point.
Standard propeller theory is not very useful for two reasons; the blades are cascades rather than isolated airfoils (i.e. the distance between blades is only a few chords, so flow is essentially parallel between blades), and the fan is in a duct which inhibits radial flow around the tips. This leads to an aerodynamic design considerably different from a propeller.
The shock waves generated at the tips propagate non-linearly and generate the rasping 'buzz-saw' noise familiar on some types - most noticeable when the tips have only just gone sonic.