Carnivores probably have a higher incidence of parasites, but the biggest hazard comes from biomagnification.
Biomagnification has its roots in the energy loss between trophic levels in the food chain. Plants are at the very bottom - they get their energy from the sun, which, for the most part, is inexhaustible. The next level up is consumers, like cows, rabbits, and so on. They consume the plants. The plants use energy for their own, for growth and maintenance... which means that there is an energy loss between the sun and the consumers. The next level of consumers (the carnivores) must eat even more herbivores, because the herbivores use energy from plants to grow, move around, and reproduce... which is another chunk of energy lost.
This means that, as you go up, each level of the food chain must eat an increasing amount of mass from the level below it. Rabbits can survive on a kilogram of grass, while a fox would need 2 kilograms of rabbit, and a wolf would need 3 kilograms of fox, and so on and so forth (I just pulled the numbers out of thin air to demonstrate the concept - they're not accurate).
The problem is, organisms acquire heavy metals, radioactive compounds, and other toxins from the environment. Some of these are human-made, while many occur naturally. They mainly enter the food chain through plants (which absorb water and nutrients from the soil, along with the toxins). The plants get eaten by herbivores, which get eaten by carnivores and so on. Since each level of the food chain must eat an increasing mass of the level below it, the concentration of these toxins *increase* as you go up.
As a result, carnivores would have a higher level of heavy metals, radioactive isotopes, and carcinogenic compounds than the herbivores would have. These can't be destroyed or removed by cooking (i.e. you couldn't cook mercury out of bear meat). It's not so much of a problem if you limit your intake - eating bear meat wouldn't be hazardous, but eating a LOT of bear meat would be (or eating tissue that filters and accumulates toxins, like the bear's liver).