Of course more memory takes more power. An engineer can always calculate the difference. Heck, anyone can in this case.
Most Flash memory is made up of separate smaller memory die that are stacked to get more memory in one package.
For example, looking at NAND Flash specs sheets from Micron, we see that standby current values are:
4GB = 10-50 ua (1 die)
8GB = 20-100 ua (2 stacked die)
16GB = 40-200 ua (4 stacked die)
Clearly, the standby current must be added up per chip die in the package, and doubles every time the memory doubles.
However, with standby current measured in microamps (ua), there's not much to worry about since that's a fraction of the power being used for other parts in the device.
It would only have a noticeable effect if you planned to leave your device sitting uncharged for weeks at a time. E.g. for the 16GB above, that would use max 35mah of battery every week instead of just ~8mah max for the 4GB. Not a huge difference for a 1400mah battery.