Does closing vents in unused rooms save energy?
Usually not in a typical forced-air system. Closing vents can reduce airflow and raise duct pressure instead of making the HVAC system proportionally smaller.

Key points
Closing vents is not recommended as a DIY energy-saving strategy
It can reduce airflow and increase pressure in the duct system
Purpose-built zoning or properly designed equipment

Your furnace does not become a smaller furnace
A central system generally continues operating with the same blower and equipment capacity. Blocking supply registers changes resistance in the duct network rather than neatly shrinking the conditioned area.
Pressure finds weak spots
Higher duct pressure can increase leakage at weak connections. Reduced airflow can also affect equipment operation. DOE guidance specifically warns of pressure imbalances and stress on duct connections.
Zoning is an engineered solution
A zoned system uses dampers, controls and equipment designed to manage different areas. That is not the same thing as closing a handful of registers and hoping the central system adapts perfectly.
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