On which English 'Isle' must you never mention the name of a certain animal and why?

It's rabbits, because there were so many warrens there all their buildings were liable to subside. I've heard it recently but I can't for the life of me remember which island.

The 'Isle' is Lindisfarne or Holy Island in Northumberland. You must never say 'pig' but instead use 'Yon things' or 'articles'. Many stories about why - one concerns a crew of 4 being lost at sea after one of the fishermen tripped over a pig.

Portland, on the Dorset coast near Weymouth, is known as an island, though it is connected with the mainland by Chesil Beach. You should not mention rabbits in a Portland pub. Tradition has it that you could clear the bar.
Portland's stone quarrymen worried that their warrens would weaken the soil, and cause tons of rock and earth to collapse on them. Rabbits were considered such bad luck, that if a worker saw one run across their path as they headed for the quarry, he would turn around and go home again.

Senior Moments is correct - so powerful is the animal supposed to be (as an ill omen) that people will not even say the word. Even on the 'Portland Observatory' website under Mammals they put 'bunnies'.

Isle Of Man : rats. The islanders always refer to them as 'longtails'.

"It is an unwritten rule in Portland that you do not use the word rabbit."
http://news.bbc.co.uk/2/hi/uk_news/england/dorset/4318710.stm

Rabbits have been associated with bad luck for centuries on Portland; use of the name is still taboo—the creatures are often referred to as "Underground Mutton", "Long-Eared Furry Things" or just "bunnies". The fear of the word is believed to derive from quarry workers; they would see rabbits emerging from their burrows immediately before a rock fall and blame them for increasing the risk of dangerous, sometimes deadly, landslides. There have been cave-ins, and in one instance a crane operator died when his crane toppled on weak ground above the burrows. If a rabbit was seen in a quarry, the workers would pack up and go home for the day, until the safety of the area had been assured.
https://en.wikipedia.org/wiki/Isle_of_Portland