WASHINGTON—The Trump administration will allow refugee admissions to the U.S. to resume under a new executive order issued Tuesday, but people from 11 targeted countries won’t be allowed in unless they can show their arrival to be in the U.S. “national interest,” administration officials said Tuesday.

The result is that more than half of refugees will still be barred from admission during a fresh review period meant to assess risks posed by those 11 countries, officials said, in what some see as a de facto extension of the...