Cinnamon Bun, immune systems are different species to species. Some viruses target some receptors in the human immune system, those receptors might not be susceptible in bunnies or other animals. However, if you modify a virus genetically and make it susceptible to the receptors of bunnies then definitely yes, they will be infected.
This Covid virus is a kind of RNA virus that genetically changes so many times so it’s unpredictable(very low chance at the moment) when it could infect other animals. If you research the other flu viruses, some flu virus came from birds and genetically modified by its own to infect human or other animals.
Sorry, this is a complex immunology or virology. I just shared my tiny knowledge. Hope it helps.
But yes, I would be cautious when I am infected with any virus and bacteria while dealing with fury friends. When I had my bun, everyday I came from hospital, I always washed my hand before I touched my bun(lol I m lil paranoid).