Cork is oak, which is carcinogenic (in different amounts in different parts of the plant). Of course for some things one doesn't usually care about that because drinks such as wine are also stored in carcinogenic oak barrels and contain carcinogenic ethanol among other harmful things, so it doesn't really matter.

Also, corks promote microbiological contamination of various kinds, leading to e.g. the common failings in wine (there is actually a multitude of "cork" faults, not just one): https://en.wikipedia.org/wiki/Wine_fault

I'd prefer any metal or plastic to cork actually. Just because it is "natural" or "traditional" doesn't make it any good.