The better way to ask the question is: Does WiFi usage impact battery draw.

The answer is yes. The better the signal, the less power it will draw. The weaker the signal, the more processing and latency is induced, drawing more power (than normal) from the battery.

All external 'elements' draw power based on performance. This includes GPS and multi-mode SIM Card units. Put another way, turning off Wi-Fi, GPS and using only one Carrier service mode can extend battery power by as much as 75% depending on the type of unit. Batteries of 1500 mA or less are affected faster than those that have higher capacity.

Some WiFi chipsets draw more power than others. Qualcomm's SnapDragon chipset is remarkably good, but about average in battery draw. It is very easy to abuse power. It's new 60 GHz chipset for 802.11 ad standard is going to draw more power than traditional WiFi. But of course, it is also going to deliver Gig-E connectivity speeds.

Another impediment is the quality of the signal a device is receiving. The better the broadcast quality is, the easier it is for the device to process which does save battery power. The differences are minute, but measurable.

The amount of power available does affect WiFi performance. The less power there is available, the potential for data loss increases. Most smart phones do not distinguish or prioritize battery power for voice over data. Some are device configurable (power saving modes), that can auto shut off some features - including WiFi.