I can think of two issues that eating caviar might cause:
Because the 2,2 percent salt solution in which the caviar is cured does not prevent Listeria monocytogenes infection, which causes abortion, no Listeria was found when the caviar was held at 3 degrees Celsius. Effects of salt concentration in the water phase and storage temperature [2007, Journal of Food Science]
Natural (including PCB) [Ecotoxicol Environ Saf. 2008] Concentrations and hazards of organic a… heavy metal pollution, which currently appears to be less of a problem Bioaccumulation of heavy metals and the risk of… [Environmental Monitoring Assessment, 2013]
For normal, non-excessive caviar consumption in non-pregnant people, it appears to pose no additional risks; I’m not sure how it’s seen in pregnant women, but since caviar consumption is typically low, I wouldn’t say it’s a problem; however, since caviar consumption is typically low, perhaps they should avoid it altogether, similar to not eating large predatory fish because of their mercury content? As far as I’m aware, neither viewpoint has any supporting evidence.