Question:
If a bird created a nest on one’s home, in our case the gutter, and it may possibly cause damage to the home, may one destroy the nest?
Answer:
It is permitted for you to destroy the birds nest, as you have no obligation to provide the bird with a place to make it's nest, especially if it is causing you harm. If it is a kosher bird, you might be able to do shiluach haken.
Best wishes