When Can Congress Refuse to Seat a Duly-Elected Candidate?

Never.

In 1969, the U.S. Supreme Court ruled that the Speaker of the House could not exclude a duly-elected candidate for the U.S. House of Representatives. In Powell v. McCormack, 395 U.S. 486 (1969), the Supreme Court ruled that the House's Constitutional authority to judge the qualifications of its own members was post facto and could only be exercised via expulsion after a 2/3rd affirmative vote. In other words, the House and Senate have no discretion when deciding whether to seat a candidate who has been duly elected under state law, but may take steps to expel that member after they have been sworn-in. The Court decided that the constitutional rights of the voters and the states takes precedence over the rights of Congress until the candidate is sworn-in as a member. Although this case involved the House of Representatives, there is no indication the Court would view a candidate duly elected to the Senate any differently.