What are my legal responsibilities if I abandon a house in Indiana when the mortgage company has started a foreclosure?

You are liable to any liability injury to anyone on property, till it is transferred out of your name. Until they complete foreclosure, you are suppose to keep property insurance on it.
You should keep taxes up to date.

Even after they start, sometimes it will take a year or more to foreclose. Others times done quickly.

Legally it is YOUR house, till the foreclosure is complete

You will also still owe the balance of the mortgage.