The tradition probably stems from the age old practice of testing whether something was really solid gold or not by biting it.
Gold is a very soft metal, at least softer than tooth enamel, and if it’s fairly pure, you should be able to leave some teeth marks in it by biting it.
The practice of biting precious metals also allowed people to see if perhaps the gold object was really just gold plated, with something like lead at the center. If so, the gold plating could be scraped off with your teeth and, given the often bitten gold coins weren’t that thick, the plating tended to be fairly thin, so you didn’t have to bite too hard to discover whether it was relatively pure gold or not.
According to the Olympic medalist Summer Sanders, Olympians bite their medals mainly because the photographers incessantly ask them to until they do it.
The practice of biting precious metals also allowed people to see if perhaps the gold object was really just gold plated, with something like lead at the center.
If so, the gold plating could be scraped off with your teeth and, given the often bitten gold coins weren’t that thick, the plating tended to be fairly thin, so you didn’t have to bite too hard to discover whether it was relatively pure gold or not.
Olympic gold medals today are not made of solid gold (not since 1912, though they do have 24k gold plating). Rather, the gold medals are made mostly of sterling silver.
In fact, none of the athletes are actually trying to make marks on their medals or checking for authenticity, photographers simply won’t let it die as it makes a more “interesting” shot over an Olympic medalist simply holding their medal up next to their faces.