I have been told that; "If the Government allows additional time for delivery, the cost to the Government is the right to delivery by the date originally agreed upon. The law requires the contractor to provide consideration for the Government's giving up that right." However, I have search FAR, DFAR, and NMCARS and could not find the requirement that the Government "shall" require consideration to modify a delivery schedule, who's delinquency is at no fault of the Government. Is consideration a requirement?
While you won't find a "shall" statement related to delivery extensions within the Regulations, there are number of other sources that confirm the need for consideration in these situations. I personally like the simplicity of a statement I found in the Army's old COR Handbook, "When a time extension is granted, the Government will receive some form of consideration or equitable adjustment."
But it is probably better to answer your question with a legal perspective; looking at how the courts interpret certain contractual arrangements. Within your question, you mentioned that the law requires consideration from the contractor when the Government gives up a right. That statement is well supported by any legal dictionary that identifies the necessary elements of a binding contract and it is further exhibited in numerous case law resources, such as The Administration of Government Contracts.
In my research I came across an online contracting forum that referenced one such case, Aviation Contractor Employees, Inc. v. U.S., 945 F.2d 1568, 1573 (Fed. Cir., 1991). This excerpt does a great job of highlighting the significance of the issue at hand:
“The government does not deny that CO Bullock had general authority to bind the government with respect to the contract at hand. Its sole basis for arguing that Bullock was without authority to enter into Mod 41 is that the government received no consideration in return for agreeing to renegotiate the contract price on the basis of additional cost factors. Numerous decisions of one of this court's predecessors, the Court of Claims, indicate that government officers lack authority to enter into contracts under which the government receives nothing.”