Tax consequences with setting up S Corp in another state
I will give you a more general answer since I am not familiar with the details of your business. If your business if physically located in Florida, but managed from your home in Minnesota then it would be safe to say that your business is conducted in both states and as such would require registration in both states.
This way you would be able to deduct all the expenses incurred by the business in both states (which would include your home office deductions). For more specific advise I would need to know the nature of the business and the extent to which you will be involved in managing it from your Minnesota location.