Would the video card with higher memory eat more battery life then the unit with lower memory? Assume the systems are same in other areas. Same HD, Memory, what not, and under the same load.
In theory there would be a difference but in practice, there isn't.
In the case of actual RAM, more of it can actually be better on battery life simply because it means less work for the computer in terms of hard drive and CPU. It is a similar situation for GPUs. Having said that, there are too many other much more important variables associated with battery life for RAM or even VRAM (incorrect term but I don't care) to have any significant sway.