WASHINGTON—The FBI publicly urged President Donald Trump not to release a classified memo, escalating a simmering dispute over Republican allegations of improper surveillance in the probe examining whether Russia interfered in the 2016 election.

In a sign of the unusual public tension between the intelligence agency and the president, the Federal Bureau of Investigation issued a statement Wednesday after FBI Director Christopher Wray tried privately to persuade the White House to keep the memo under wraps. The document is...