Assuming that President Biden does sign an Executive Order to forgive student loans, would anyone have standing to challenge this action in court? Presumably those whose debt was discharged won't have standing because they didn't suffer any damages but how about Congress or some other government institution?
Not less than a full house of Congress, and perhaps Congress as a whole, might have standing. It is hard to see anyone else who would. The law of Congressional standing (the link is to a report of the Congressional Research Service, a policy research arm of the Library of Congress that does research for Congress) is quite involved and is not perfectly consistent and clear. Congress would argue that it has suffered an institutional injury as an institution and perhaps authorize someone to bring suit on its behalf via a joint resolution.
As the Court explained in Arizona State Legislature, an “institutional injury” is an injury that “scarcely zeroe[s] in on any individual member,” but rather “impact[s] all Members of Congress and both Houses . . . equally.”
There is considerable uncertainty regarding how this would be applied which is not really at issue in this case at the present time since Democrats control both houses of Congress and support the President in this policy.
Individuals legislators lack standing to sue in a case like this one.
See also Tara Leigh Grove & Neal Devins, "Congress’s (Limited) Power to Represent Itself in Court", 99 Cornell L. Rev. 571 (2014); Matthew L. Hall, "Standing of Intervenor-Defendants in Public Law Litigation", 80 Fordham L. Rev. 1539 (2012).
A blog entry from a law professor considers the question and comes up with the Congressional standing analysis above, the notion that a loan serving company paid on the dollar value of the loans serviced might have standing (which isn't inconceivable but is a stretch), and finally considers "competitor standing", a minority view that I do not think is sound in these circumstances (because the forgiveness is retroactive only and does not change competitive positions going forward).