Op-Amp Current Consumption
There are a number of curves showing the supply current for different supply voltages and for different temperatures and common-mode voltages, so you can predict the typical supply current. It increases with increasing supply voltage, but fairly gently.
As far your your load current, you'll need enough swing to satisfy your actual requirements, and again the datasheet provides a great number of curves. Once you have enough voltage to (worst-case) supply your required output swing, then additional voltage simply results in more power dissipation in the chip, since the additional output current is drawn directly from the power supply terminals (and possibly a bit more).
The total current drawn is the sum of the two, so it's Is + k\$\cdot\$Iout(Vsupply) where k may be a bit higher than 1, and this the power dissipation in the chip is (Is \$\cdot\$ Vsupply) + k\$\cdot\$Iout(Vsupply-Vout).