Is it possible to update the firmware of a camera (like Nikon D810) to enable 4K video? Or, such features are built into the hardware?
Yes and no. Any camera with a sensor larger than 8mp (and the correct aspect ratio) is capable of recording 4k video frames. Firmware can grab frames off the sensor as fast as the sensor can read out. The ability to record video at a given frame rate is a limitation of the sensor's ability to read fast enough, the processor's ability to encode fast enough and the card reader that can write the data out fast enough.
Practically however, most cameras that don't have 4k are unlikely to have fast enough performance in all these areas to be able to hack 4k resolution in. It's simply too much data to move and most cameras didn't design that kind of over-engineering to be able to deal with all 3 of those conditions. Even the Canon 5D Mark iii, which is a high end professional DSLR which can be hacked to shoot RAW video at decent frame rates at 2 to 2.5k ends up having to be dropped down to around 12 or fewer FPS for trying to hack it in to shooting 3k or 4k video.
This is also partly compounded by the fact that most of the real time encoders are built for particular resolutions. For example, the previously mentioned 5D Mark iii hack works by recording raw frames which takes a huge amount of data rate for storage.
So, in some cases, there may be ways to get some level of 4k support from a third party firmware, but for the most part, you are limited by what hardware was originally put in and often (most of the time) it simply isn't going to be possible to increase to 4k by third party hacks.
As for the Nikon D810 specifically, I'm not aware of any highly developed third party firmwares for Nikon bodies that are similar to Magic Lantern, so I don't think there is anyone really exploring that in depth at this time for Nikon bodies. That series is similar in spec to the 5D mark iii, so it may be possible to get low framerate 4k video on very fast cards, but that's purely speculation. I wouldn't count on seeing it any time soon or having it be highly usable though.