Check your original loan offer, at one stage banks were offering same rates to home loans and investment properties so there was no issue with whether or not the status of a house changed as all rates were the same so it depends when you got your mortgage. There may be no mention of that stipulation in your loan offer, if there is not then I would asking them on what basis they were changing your rate.
If you get a mortgage elsewhere you are under no obligation to tell them. However, if you rent your original house you will have to change your house insurance and if you have to send them a copy of your new policy that may alert them as obviously it will be for a rented property. Although some times a letter of indemnity is issued by the insurance company and the bank accepts this, usually states just address and amount of insurance. Is your buildings insurance through the bank at the moment?