Does home owner's insurance cover a leaky drain pipe from a bathtub that causes water damage to the walls?

If you have "accidental water discharge" coverage on your policy then it will cover damage that results from a water overflow or a leak. If you do not have the coverage then it will not be covered.

Your policy will not cover the cost of the plumber to fix the leak if it's a normal maintenance issue.

You will want to call a plumber first and get the leak fixed before the company will pay for any resulting other damages.