What happens when a person who is willed a house and has to buy the house fails to close on the house by a certain date?

I am not sure I understand your question.  I assume there is a will that transfers the house to a named beneficiary.  Does the will provide that the beneficiary must buy the house from the estate, or is there a mortgage on the house and the beneficiary must "buy" the house to release the mortgage?  Does the will provide a date certain by which the beneficiary must close on a sale?

If the will provides for all these contingencies, then the will tells you what happens when the beneficiary fails to close by that certain date.  If not, the consequences will depend on the executor's decision and the court's approval.  In general, courts do not like to void a gift in a will.  Courts will usually allow more time to close the sale.  If the beneficiary cannot buy the house, the court will try to achieve the testator's intent some other way.

I suggest you call the lawyer who is handling the estate and ask what happens next.