What Does the Constitution Say About House Impeachment Proceedings
Short Answer: Essentially nothing
I've seen a lot of learned commentators on both sides of the impeachment debate arguing that the House *must* follow certain procedures (or not), or that the president *must* cooperate in the following ways (or not). What almost no one ever does is quote the relevant constitutional text, which is ridiculously sparse: "The House of Representatives shall chuse their Speaker and other Officers; and shall have the sole Power of Impeachment." There is nothing about what procedures the House must or may use, nor is there any indication to what extent the president and other executive branch officials are required to cooperate.
What we do have is historical literature on what impeachment was thought to be in 1789, analogies to other civil, quasi-criminal, and criminal proceedings, past practice by both the executive branch and the House, and any relevant Supreme Court precedents on related matters (even though most impeachment disputes will not be justiciable, all parties still have to fulfill their constitutional obligations, which judicial precedent might speak to.)
But how, for example, would one weigh executive refusal to cooperate if, for example, historical practice, the "best" analogy, and SCOTUS precedent all provide different answers? Color me skeptical that there will generally be definitive answers to the questions raised by impeachment other than however the political process sorts things out.
UPDATE: What about this language from the Constitution? "Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behavior, and, with the Concurrence of two thirds, expel a member."
Some issues: (1) It's not clear that the impeachment process is an ordinary "proceeding." After all, a Senate trial is presided over by the Chief Justice, so know there are outside limits over the trial via the Chief; (2) The Supreme Court found that the authority granted in this language is not absolute when it prohibited the House from expelling Rep. Adam Clayton Powell; (3) Even if the House can make its own rules, one can argue that when it's acting as a quasi-judicial body, it has an obligation to follow its own procedural precedents, just as actual judicial bodies do (even if that obligation is not justiciable in the courts); (4) This begs the question as to whether the House has in fact established any rules for the current proceedings, given that there has been no formal vote to start impeachment proceedings; and (5) Regardless, it still doesn't tell us under what circumstances it's legitimate for the executive branch to resist cooperation with impeachment proceedings.