The biggest enemy of pigments is ultra-violet and lots of cheap flash kits do not filter that out.
Sunlight is about 1 KW/square meter on a non-cloudy day at noon, directly from above.
4% of that is a typical value for the UV component.
So the UV power hitting a 1 square meter surface at noon is about 40 Watts.
Photographic flash is on the order of a millisecond give or take, and power output of a typical flash during that millisecond is 100 Ws, or 100 KW/ms. Depending on how far away you are from the work of art you're putting for every 10 flashes then the equivalent of about of a second of exposure to the sun on the totality of the work (much more than 1 square meter, so you have to divide by the total surface). In the case of the chapel all the light discharged in the chapel will hit the object somewhere so apart from atmospheric effects (likely small) you can assume all of that light would contribute to the fading.
So every 10,000 tourists passing through and making one picture each with flash on is the equivalent of about 1000 second's worth of exposure to direct sunlight (think of a window of one square meter opening for 1000 seconds once per day), all this assuming that the flash contains an equivalent amount of UV as sunlight which likely is not the case. About 20,000 tourists per day pass through during the peak of the season, so on such days that's the equivalent of 2000 such seconds of the open window.
So at a guess, this is probably not too much of a problem, unless the pigments used break down very easily. (Blue is usually the most stable, with other pigments being more susceptible to fading)
What would be a problem is the fact that if all these people are discharging their flashes all the time that this seriously interferes with the ability of others who do not engage in such annoying reproductive behaviour to appreciate what they are looking at.
edit: fixed some of the math