The move to separate the Philippines from the United States has already seen a burst of anti-American sentiment on the streets of the country. It is likely that this will be a moment of historical myth making for people in the Philippines and it is likely that they will become more vitriolic in their distaste of the United States. A cooling of relations seems inevitable.
President Duterte's announcement that the Philipeans should seperate from the US is going to have an incredibly huge effect on how the two countries relate to each other in the future. It will take years, for the relationship between the two countries to be repaired if it ever can be.
While the Philippine President's announcement of a separation from the United States may suggest a permanent harm in relations between the two nations, I do not believe that such a decision will necessarily have a lasting effect. The future is uncertain, and social, political or financial occurrences could arise that would require the two nations to reach a level of cooperation once again.
No, this will not permanently harm relations between the two countries because Duterte cannot be in office forever. Once a new leader replaces him, the US will be able to work with the country again. Duterte has insulted everyone, regardless of where they are from and should be removed quickly.