Probably one of the worst things tuna has, is the high levels of mercury. Even in humans, high levels of mercury acts as a neurotoxin, damaging the brain and nervous system. Since cats have a much smaller body mass than humans, their tolerance for mercury is going to be lower as well. For that reason, it's important to limit how much tuna you feed your cats.
There was actually a study done in 1974 titled "Neurological Changes in Cats following Long-Term Diet of Mercury Contaminated Tuna" by Louis W. Chang, Seiya Yamaguchi and Alden W. Dudley, Jr.
Unfortunately the article is behind a paywall, but it can be accessed here, and the first page and an abstract can be viewed. I will try to summarize my understanding of the study below:
Sixteen kittens (both male and female) were fed a daily diet of tuna
that contained a mercury content of 0.5 parts per million. They were
given vitamin supplements to cover for the nutrients lost in a tuna
only diet. The study lasted 11 months and an average of 21.1kg of tuna
was consumed by each cat, along with an average of 6.3mg of mercury.
What they found was that there were no observable changes in their
growth rate or body weight, but at seven months three of the cats
showed outward signs of neurological disturbances. Of the three cats
showing symptoms, two of them showed transient symptoms (meaning the
symptoms seemed to disappear after 24 hours), and the third displayed
severe ataxia and incoordination in movements until the end of the
After the 11 months, the insides of the cats were observed, and
histopathological lesions were found in the nervous systems of the
majority of the cats studied (I couldn't find the exact number). The
most prominent lesions were found in the cerebellum, at the deep sulci
of the cerebellar folia. Many of the granular cells (neurons) were
found to be dense and pyknotic (Irreversible condensation of cells).
The amounts of mercury allowed in tuna hasn't changed over the years I don't think. Even today, the amount of mercury found in Tuna, on average, matches the study at 0.5ppm. But it is important to note that the EPA does not take action until it reaches 1.0ppm, which means that you're likely to see ranges both aboth and below 0.5pmm to reach that average. The FDA has a monitoring program as well.
There is another dangerous chemical that you can find in canned tuna, and raw fish in general. That chemical is thiaminase. Thiaminase is an enzyme that attacks and inhibits the processing of thiamine, commonly known as vitamin B1. Cat's need high contents of B vitamins in their diet, so feeding a cat too much raw fish would give them a thiamine deficiency. The side-effects of which include seizures and a loss of control over body movements.
It's important to watch what the tuna is packed with as well. Tuna comes packed with either water, or oil. The oil is bad for cats as it can cause a vitamin E deficiency, which in turn causes problems with their muscles1. So you'll definitely want to try and stick only to tuna packed in water.
Check the ingredients as well. Tuna also comes either salted or unsalted. As too much salt can cause electrolyte imbalances in cats, it would be best to only get the unsalted tuna (You can always add salt to the portions you eat). Some tuna also has onions and garlic added to it. Both are poisonous to cats.
Other things that I would suggest you watch for is if your cat has possible allergies to fish, as fish is one of the more popular allergies that cats can have, and if your cat becomes to addicted to the taste of the fish. Just like children, give them a choice between something delicious tasting and something healthy, they'll take the delicious tasting food.
With all this being said. It's won't be harmful to give your cat(s) a tiny bit of tuna as a treat on occasion. But it's really important to keep it in moderation.
There are some good uses for tuna. Because it smells so strongly, it can be used to capture stray cats for catch and release programs. If a cat is sick and/or refusing to eat, a little bit of tuna mixed in might get them to eat.
1See "The Cornell Book of Cats: A Comprehensive & Authoritative Medical Reference for Every Cat & Kitten" and "Nutrient Requirements of Dogs and Cats (Nutrient Requirements of Domestic Animals)" for more information on Steatitis.