Shift register 5V VCC and data while 3v3 serial?
To directly answer your question: using 5 V Vcc and 3.3 data and clock will not damage your shift registers, but may result in unreliable operation, if the 3.3 volt signals don't meet the Logic High spec for the shift registers.  You can avoid this problem by using the 74HCT (TTL-compatible) versions of the shift registers, as they have lower Input High thresholds.

I don't understand your concern about putting a load on the microcontroller if you use a 3.3 volt supply.  If you have an available 5 volt supply, you can use a 3.3 volt regulator from that supply to provide 3.3 volts to your shift register, if you are concerned about overloading a 3.3 volt regulator on you microprocessor board.