Traducción:Este tocador va a ser demasiado pesado.
I think either ser or estar could be OK. For example: Ser...the dresser itself, even without anything in it, weighs too much for us to be able to carry it up the stairs. In this case, ser is appropriate because it is an inherent characteristic of the dresser itself...it was made in such a way that it weighs a lot.
Estar...once the dresser is full of all of the stuff that we will need to put in it, it will be too heavy for the floor where we need to put it. In this case, estar is appropriate because the weight of the dresser can vary from day to day or month to month based on its contents...and it is fine in Spanish, as in English to refer to the weight of a piece of furniture based on its status...empty vs. full, etc.
One of the difficulties of Duolingo is the lack of context...by reading the comments, however, we can learn a lot about this aspect.