US CBP ordinarily won't care because they are not generally tasked with enforcing the other country's immigration rules and because a valid US passport is valid in their eyes for the entire period of validity noted in the passport (unless other action has been taken by the US government to invalidate the passport).

The airline more often than not won't make an issue of it, but they ought to pay attention to the passport validity date-related rules (if any) of the receiving country.

When entering the UK, the UK immigration authorities haven't cared to stop entries on my US passport on the basis of upcoming passport expiration date, but then again I've always had plans to depart long before the passport would become invalid for travel.