When driving a Nixie tube with the HV5622 shift register, how hot will the chip get?
Most of the power dissipation will be from the Vdd current in your application. If you run it at 12V Vdd, say, note that it can draw as much as 15mA, which is 180mW while clocking it at 8MHz, according to the datasheet. It's not clear (to me anyway) what the Vdd current draw is otherwise or whether there is a preferable state for the input lines. The datasheet is not very helpful, I'm afraid. For example it refers to Vin = 0, but there is no pin called Vin. It's possible the dissipation under static conditions is worse than 180mW but I would guess not (and verify!). 



Edit: I looked at a similar datasheet from the original manufacturer (Supertex, who was acquired by Microchip) and Vin seems to refer to the output voltage, which is .. weird.  



The output voltage is guaranteed to be less than 15V @ 100mA out and your output current will be perhaps 3mA per output. Probably the output voltage will be well under 1V, so allow less than 10mW for all 3 outputs. 

There is also a bit of dissipation from leakage currents but that won't normally be significant.