How does the current draw of an IC change with supply voltage?
It depends on the IC.

For ordinary digital chips, the dominant current draw is charging and discharging capacitances to full rail voltages, so current draw is likely to be roughly proportional to voltage. 

For analog chips the current is also likely to be positively correlated to the voltage, but the correlation may be weaker than with digital parts as analog designs are often based around (roughly) constant currents derived from some reference.

And very occasionally you will come across a device with a built-in switched mode power supply so the current drops as the voltage increases.