Assuming that they would continue to stand there if the Court ruled that the occupant in that office was not eligible to be there. The Secret Service is sworn to protect The President, and the Marine is likewise obligated to follow legals orders only, and is likewise obligated to protect The President. If the person in the Oval Office is only the occupant, and not the President, why would they protect him/her? They might even aide in tossing him/her out, so that the real President, whoever that might be, can take his lawful place.
It's all unknown territory.
But I don't think they would order federal marshals, who are part of the executive branch anyway, to remove an ineligible person. They would just rule on that eligibility. At least unless or until asked, they likely would not even order him to vacate the office.
< But, as the Court has said, an unconstitutional law is unconstitutional from the time it is passed, not just after being declared so. Similarly, a person ruled ineligible to the office, can never have actually held it or legitimately exercised any of its powers.
Moreover, there is ample legal precedent for the Supreme Court to take this position and quite properly so. After all, the matter could have been handled by the political parties themselves, by the secretaries of state of the individual states which certified candidates or ballots, or when they certified the election results, or by the political party in convention, or by the House of Representatives and the Senate. The Supreme Court is not the only venue for resolving disputes in America and it is not the venue for deciding "political" questions at all.
Just because all those other folks did not exercise their "due diligence" doesn't mean that the Court should not enforce the Constitution, or rule on a particular case or controversy arising under the Constitution. Besides none of those folks even examined the eligibility of the candidates or the electee. (Except perhaps for the parties who nominated him, who declared him eligible to various State officials).
In the end, it is not a "political question", involving some policy matter, but rather it is a Constitutional question of the first magnitude.