Assuming that any dynamic backlighting features are disabled, a typical TN panel will use slightly LESS power on an all-white screen than on an all-black screen.
LCD panels comes in more than one flavor. The cheap and common variety is called TN (Twisted Nematic). A TN pixel is transparent/white until power is applied to it, at which point it turns opaque/black. This means that it takes some small amount of power to make a pixel turn black. A less common, more expensive technology is called IPS (In-Plane Switching). An IPS pixel is opaque/black until power is applied, at which point it turns transparent/while. There are also VA (Vertical Alignment) panels, and I don't remember off the top of my head whether they're transparent or opaque when not powered.
I just ran a test with a power meter of my own, and the IPS-based LCD monitor I'm using consumes 17 watts when displaying an all-black image and 21 watts when displaying an all-white image, and the backlight is not responsible for any of that difference. Tests I've done on a previous TN-based LCD monitor have shown an opposite result, as is to be expected. The thing is that TN panels are far more common because they're cheaper to produce. So if the entire world tries to use use dark web pages in order to save power, then a few people with IPS panels will use less, but the vast majority of users will be be browsing on TN panels, and these users will actually be using slightly MORE power. And, given their greater numbers, I suspect the average power usage would actually go up.
And as long as there are any white or brightly colored pixels onscreen, be they part of the taskbar, the browser's title bar, or the web page itself, then dynamic backlighting shouldn't be much of a factor, either. The backlight can't be turned down any dimmer than the brightest pixel. (This excludes LCD monitors with backlights that are capable of zone-dimming, but those are exceedingly rare.)