Is it reasonably possible for a type of human to exist that rapidly tans in the presence of large amounts of sun/radiation and loses it again afterwards, in a matter of minutes? Also, how might such a mechanism exist and how could this affect the body in other ways?
Rapid tanning is beneficial
My skin darkens really fast. One sunny day, I get reddish brown. Next day I have a good tan. I can go with sunscreen filters as low as 6 and don't get burnt. In an early history of our species it would mean that I'm not incapacitated by sunburns at the first really sunny day. I would eat, and my children would eat.
My friend needs sunscreen as high as 50 and he still has blisters anyway. He needs a few weeks to develop tan and even when he reaches his max, his protection isn't nearly as good as mine. Without technology, it would be a huge evolutionary drawback. On a sunny day he would struggle to feed his family.
It is fast enough now
In people who are tanning the fastest, sunburns are next to nonexistent. There is no evolutionary pressure to go even faster. And there wasn't. You would need some in your world for this to happen. A bright UV source that turns on or off suddenly would do. Or unavoidable radiation if you want other pigments, to absorb other wavelengths, to evolve.
Rapid loss of pigments is not useful
It is costly for organisms to create (and replenish) pigments. That's why only people who originated near the equator have dark skin all the time. Decomposition of these pigments serves no purpose. They decompose naturally and are not replenished if not needed, but making this faster means that the body wound need to redo the same work the next day. That's a waste, when pigments may last for weeks. Thus, there is an evolutionary pressure against speeding up this process, and more pressure to speed up tanning, the more pressure to slow down brightening.
There are creatures which are able to rapidly change the pigmentation of their skin cells, in order to adjust to the environment. Examples are octopusses and cuttlefishes.
Since tanning is yet another pigmentation of the skin, in principle it could be possible to have dedicated cells tuning the melanine content of the skin outer layers according to radiation intensity.
The only problem I see with such mechanism is that it doesn't bring a clear evolutionary advantage. In a shaded enviroment the skin would turn pale and the individual would appear whitish, thus more visible than a dark skinned individual. On the other side in a fully sunny place the individual would appear dark brown, again more clearly visible than a pale one on a light colored background. Therefore a predator would have higher chances of spotting the pray and kill him/her. On the other hand, skin cancer as consequence of UV radiation kills in few years, way longer after the human would have been mauled.
What you are talking about isn't actually tanning (though it may look like it to the casual observer), but a rapid pigmentation change in response to light and/or radiation.
Take a look at the Photochromic Lens. Basically these are the lenses in glasses that darken and lighten depending on the light level.
Typically, photochromic lenses darken substantially in response to UV light in less than one minute, and continue to darken a little more over the next fifteen minutes.1 The lenses begin to clear in the absence of UV light, and will be noticeably lighter within two minutes, mostly clear within five minutes, and fully back to their non-exposed state in about fifteen minutes.
There are two different versions of photochromic material used in these lenses. You would likely want to go with the more biologic version oxyzines or Naphthopyran.
There are animals that change color but this is for camouflage. Those mechanisms are caused by
xanthophores, which contain yellow-red pigments iridophores containing colourless stacks of crystals or platelets that reflect and scatter light to generate hues such as blues, white and ultra-violet
melanophores, which contain black melanin pigment
Rapid colour change may occur due to various "triggers" including temperature or light (a reflexive response via light-sensitive receptors in skin). source
You'd want to look at these more than you'd want to look at what ordinary humans have for such a mechanism in your species.
As such, I would say that you'd have to make a new species. You can use human as the template, but the genetics of such a species will be different than ordinary humans.
I am with L.Dutch when it comes to it being an evolutionary disadvantage, however,
chameleons are very pale at night when asleep but darken as soon as a torch is shone on them (and only on the side with the light shining on it).
If your humans worked this way, they would also have to have an advantage. And there advantages to color changes in the wild. Cold blooded animals turn darker in the sun to absorb heat and when they are too hot, they can turn pale to reflect heat. It's also used for camo and communication.
If your planet had more radiation or a different kind where this protection is more of an advantage than constant melanin, of course it would be helpful evolutionarily. But if the cost of the change is not as simple or advantageous as melanin, then of course, it would not happen.
Also, how might such a mechanism exist and how could this affect the body in other ways?
Well, take a look at the advantages that color change affords other species, and specifically what they are. Mammals don't generally, and if they do, it's in response to a season and happens over time, and mostly it's fur and has a lot to do with temperature. Like a Siamese cat--their pattern of fur color is dependant on their body temperature over time! As older cats have less circulation and are less warm, they get darker as they age. So, depending on what track you go, this can impact the genes of your interesting humans. You're talking about a far more rapid change in color, which might lead to something more...reptilian, but it's worth looking at these nifty cats.