In theory, they could also invoke executive privilege. Simply refusing to answer the subpoena without even asserting an excuse would likely constitute obstruction, but there’s no self-executing remedy–one “remedy” could be a contempt vote. However, we’re now in impeachment-land, and the more likely route is, as with Nixon, the failure to cooperate with the investigation is added as an article of impeachment and is taken into consideration when voting to impeach and, in the Senate, to convict.
The executive branch knows that Congress is no longer messing around. If they stall and delay and assert BS privileges or refuse to answer questions or answer falsely, it just hurts their claim that they shouldn’t be impeached and convicted. Even most Republicans in the Senate care that the Executive not place itself above Congress’ constitutional authority. Note that Clinton actually “cooperated” with the impeachment proceedings and Trump will be expected to do the same (lol), and will have very little in the way “legal” arguments not to.
Remember, Nixon was ordered to hand over the tapes–of highly “executive privileged” communications within the oval office with his top advisors–because privilege is actually very narrow and doesn’t attach to crimes. In this case, there’s no time to go to the SCt (and Dems don’t want to), and (just like with failure to produce relevant evidence in a civil trial), any failure to cooperate with the impeachment by producing witnesses and evidence will give rise to adverse inferences that the material or testimony would be damning.
Congress has been playing (poorly) according to one set of traditional rules based on norms and accommodation between two co-equal branches of government. However, the Executive is now the defendant in a quasi-criminal inquiry, and the rules are now different. The gloves are off. He can simply stonewall, but the House will vote to impeach, and significant stonewalling will genuinely anger the Senate and, at best, significantly damage Trump’s already weak reelection prospects.