It is your responsibility to tell the mortgage company of any changes like this. The insurance company wont do this for you unless they are required to send a copy of the policy, and then the lender may not read it.
When told, some lenders used to just say "thank you for telling us" and leave it at that. Others will want to change the terms of the mortgage - this could involve a higher monthly payment, a one-off fee or, as you say, a payment to reduce the LTV ratio. It's a good idea to talk to them before you make the final decision about renting so that you can take account of the implications.