In a statement  on Sunday regarding President Trump's moratorium on immigration from several majority-Muslim countries, Senate Minority Leader Chuck  Schumer described the policy as "unconstitutional" and said (emphasis mine):

The refugee program should be for all religions that are persecuted. That's the point. There should not be a religious test. The Constitution prohibits it.

Let's parse that carefully.  The Constitution does, in fact, prohibit "religious tests" – but does it prohibit religious tests for foreign refugees?  The relevant passage is Article VI, paragraph three:  

The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.

Clearly, the "religious test" clause prohibits religious tests as a condition for elected or appointed officials to hold office.  It absolutely does not apply to any immigration policy or refugee policy.

Chuck Schumer is a Harvard Law School grad.  He was first elected to the Senate in 1998, and he served in the House of Representatives before that.  With a résumé like that, Schumer ought to know the Constitution well enough to know that his statement is completely false.

Is Schumer constitutionally illiterate…or is he deliberately misrepresenting the Constitution to deceive the public?

I'd bet my money on the latter.