Only if it gets killed by the player or hostile mobs.
This of course would not include killing the golems with lava etc.
The Villagers would be without a protector and so they would be too afraid to walk around their village unprotected from all kinds of dangers. Eventually they would overcome their fears and start spreading the gossip about the need of a new Iron Golem.
[EDIT]: Since 1.14.3pre1 just dropped, remove the gossip spreading part
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