It is always your responsibility as a tenant to take good care of the property you are renting, including keeping it clean and preventing damages. This includes tasks like removing limescale from fixtures, cleaning drains, maintaining the range hood, and ventilating to prevent moisture damage and blockages.
You are also obligated to inform the landlord about any damages and faulty fixtures, whether they are due to negligence or regular wear and tear.
In the Danish Rental Law (Lejeloven), a distinction is made between interior and exterior maintenance responsibility. This refers to specific areas and parts of the rental property and who is responsible for maintaining them.
Your lease contract will specify who is responsible for maintenance and what it entails for you.