Do constant current LED drivers consume significant current when all outputs are disabled?
In your example, BLANK is not an &quot;enable&quot; pin.  It turns all outputs off, or allows the on/off states as set in the registers.  It does not enable or disable the rest of the chip.
Since everything else is enabled, the logic sections of the chip and the reference current will still be active.
The logic sections don't seem to consume much current, but the reference current can.
Depending on how bright you've set the output (smaller resistor for Rref,) the higher the reference current will be.  That current is consumed even when the lights are all off.
In the table, there are four examples with various logic states and reference current resistor values.
The current ranges from 1mA (dim lights) to 40mA.  That tracks very well with the reference current as given in the chart on page 13 of the datasheet.  That chart is the output current, but the output current tracks the reference current.
The chip itself consumes a few milliamperes in addition to the reference current.
Other ICs may work differently.  In any case, BLANK does not shut off this IC.