No. The landlord does not have an obligation to paint between tenants.
However, the landlord must ensure that walls and ceilings are in good repair under the Minimum Housing and Health Standards. This includes no cracks or holes and that they are easy to clean. Bathroom walls must also be smooth and non-absorbent. Kitchen walls must in very good condition to ensure that bacteria and pests do not become a problem. These requirements mean that while a landlord does not have to paint between every tenant, the landlord should paint regularly to meet health standards.