Will playing a 1080p movie drain battery faster than a 720p movie?

This seems like the type of thing where one experiment trumps a thousand expert-guesses-- even from the engineers who designed the phone. Hopefully someone else will answer you definitively, but in case nobody does, as an electronics engineer who has worked in the graphics industry, I would guess that...
1) If you're displaying the 720p video as 720p (in a black letter box) on your phone's screen without scaling it to full-screen, then I'd guess the 720p video will use less power because the video decoding is less compute-intensive.
2) If you're displaying the 720p video scaled up to "full" screen (1280 pixels wide, cropped or 1080 pixels wide) then I'd guess that there will be little difference in battery usage (compared to 1080p) any power savings from decoding 720p would be spent scaling the 720p video up to 1080x 720 pixels.

However, there are other critical factors that you didn't mention, like bit-rate, file format & CODEC (MP4, AVI, MOV, MPEG2, WMV, etc.), and streaming vs playing from SD-card, but I'm assuming that you meant for the same bit-rate with the same file-format, and played from the SD card.

If you're talking about streaming then I'd guess that 1080p will take more juice no matter what.

If you figure out the answer for certain, I'd curious to know.

 

Nope. Whatever color wallpaper you use, the LCD (any type) screen will still turn on all the backlights. Darker wallpaper to save power only works on OLED (whatever type screen that has OLED in the name) screens because each pixel in an AMOLED screen is basically a tiny light bulb (actually 3 bulbs/colors per pixel). A dark or black image would mean more bulbs turned off, hence saving battery.

 

Turning down the brightness is the only way to save battery life for LCD screens.