I asked this question at law.stackexchange.com where it was closed as opinion-based.
From the New York Times:
[...] Sean Spicer, the White House press secretary, barred journalists from The New York Times and several other news organizations from attending his daily briefing, a highly unusual breach of relations between the White House and its press corps.
[...] Reporters from The Times, BuzzFeed News, CNN, The Los Angeles Times, Politico, the BBC and The Huffington Post were among those shut out of the briefing.
Have First Amendment rights been violated?
Nothing from Cornell Law's page seems to indicate that not admitting certain parties to a press conference is in violation of the First Amendment.