The phrase "builds fire under" means to create urgency or motivation in a situation. The antonyms for this phrase are words that represent the opposite meaning such as calm, discourage, or demotivate. Calming down a situation is an antonym of building fire under it. Rather than trying to encourage someone or make them move faster, a person who is discouraged or demotivated is unlikely to take any action. Other antonyms for "builds fire under" could include delay, procrastinate, rationalize or just put things off. These are all actions that will slow down or stop the flames of urgency from building.