It's a topic I've often mentioned, usually regurgitating tests done on Symbian-based Nokias from the late 2000s - that using a dark theme/applications can save battery power, especially on AMOLED-screened devices. But it's great to have some data points in the Windows Phone world, in this case via 7tutorials. Quotes and conclusions below....

From the article:

We took the following smartphones with Windows Phone:

• Nokia Lumia 525 - it has a 4” IPS LCD screen, with a resolution of 480x800 pixels.
• Nokia Lumia 625 - it has a 4.7" IPS LCD screen, with a resolution of 480x800 pixels.
• Nokia Lumia 720 - it has a 4.3" WVGA IPS screen, with a resolution of 480x800 pixels.
• Lumia 1020 - it has a 4.5" AMOLED RGBG PenTile ClearBlack screen, with a resolution of 1280 × 768 pixels.

...We used the default configuration for each smartphone and the default colors that were set by Nokia. Then, we added a Microsoft account and installed the WP Bench app, so that we could test the battery. Also, we made sure that each smartphone had the latest version of Windows Phone that was available for it.

We changed the background color from dark to light and ran 3 battery tests on each smartphone, for each background color. Then, we calculated the average score for each smartphone and each background color and we looked for any obvious trends in our measurements.

Below you can see the average time each smartphone lasted in the WP Bench battery test. This test stresses the phone to the maximum and it drains the battery really fast. In real life your smartphone’s battery won’t drain that fast.

If there’s one thing clear from the chart below is that you will save battery when using a dark background instead of a light one.

You can read the full article here. Of note is that the WP Bench utility adopts the current theme (i.e. dark/light) for its main battery test, making it a good choice for doing this kind of intensive test. 

On the slower Symbian phones of 2007, the screen output was much more significant. On today's multi-GigaHertz processors and chipsets, and especially when running an intensive battery-draining utility, as here, the fraction of battery power needed by the display is comparatively less, but there's still a significant difference between dark and light themes on the AMOLED-using Lumia 1020, albeit only about 8%.

What's perhaps surprising is that LCD-screened devices also show savings and I can only think that the display electronics is intelligently managing the backlight power according to content, with the effect largest on the Lumia 720 and smallest on the small-screened Lumia 525, as you might expect.

Either way though, if you need an extra 5% battery to get you through the day, think about switching to a dark theme - every little helps!