The natural human cornea is opaque to shorter-wavelength UV, but there's no reason it couldn't be replaced with something transparent to UV-A at least. This is supposedly an unintended side-effect of some cornea replacement surgery[1].
UV may have the potential to be damaging, but plenty of animals see UV-A safely - for example, bees[2] and mantis shrimp[3] can perceive UV to 300nm. Some birds (whose eyes are much more like ours, of course) can also perceive UV[4][5].
If we wanted some ready-made genes, I vote for borrowing from the European starling, which has receptors that sit in our enormous gap between blue/violet and green as well as UV-A [6].
Near infrared could be very useful, because (in combination with red, green, etc.) it can be used to easily distinguish vegetation, and it scatters less in the atmosphere, so it could allow clear vision at much farther distances. Most materials that are transparent to red/green/blue are also transparent to NIR, so as far as I know, the rest of the eye shouldn't need to be modified.
Most people can apparently see a little bit into the NIR if they wear NIR-bandpass goggles, but it's extremely dim and doesn't have a perceived colour that's distinct from the rest of the visible spectrum, so that's not really what the GP is suggesting. They're suggesting wiring up a true NIR-specific receptor in the eye which would provide a fourth, fifth, or sixth primary colour (in addition to red, green, blue, and optionally the in-between-red-and-green that tetrachromats see as well as a hypothetical UV-A receptor).
I can't find a cite at the moment, but some birds have NIR patterning on their feathers (I have photographed this myself at a zoo), and the last time I looked into it, there was some speculation that tropical birds in particular had some NIR sensitivity (in addition to UV-A) and had evolved spectrally-complex colouration as a result.
Thermal infrared is a much longer wavelength, and I'm not aware of a good material for lenses/eyes that is transparent to both. It's possible to have an uncooled sensor that detects it, but I agree that it's a much larger leap than UV-A or NIR.
[1] http://www.itworld.com/article/2732596/consumer-tech-science...
[2] https://www.agriculture.purdue.edu/agcomm/newscolumns/archiv...
[3] http://www.iflscience.com/plants-and-animals/mantis-shrimp-s...
[4] http://www.livescience.com/26994-how-birds-uv-vision.html
[5] http://www.webexhibits.org/causesofcolor/17B.html