Is furnishing the marital home supposed to be part of the mahr (dowry)?

Praise be to Allah.

The mahr is the exclusive right of the women, to dispose of as she wishes. She does not have to prepare the house, because there is no text in the sources of sharee’ah which state that the wife is under any obligation to prepare the marital home. There is nothing to state that her parents are obliged to prepare the house either. No one has the right to force her to do that. If she does anything to prepare the house or the furnishings and equipment that are needed, this is voluntary on her part. 

Preparing the house is an obligation on the husband. He is the one who has to do everything that is needed to prepare the marital home by supplying furnishings, conveniences, equipment and whatever else is needed in the home, because that is all part of the spending which he is obliged to spend on his wife.