This is a good resource on the subject of Executive Privilege
Highlights from the recent court decisions around subject:
With regard to the executive’s claim, the court noted that there was no absolute claim
of executive privilege in response to congressional requests even in the area of national security...Congress’ investigatory power is not, itself, absolute...According to the court, judicial intervention in executive privilege disputes between the political branches is improper unless the branches have made a good faith effort at compromise without result. The court held that there is a constitutional duty for the executive and Congress to attempt to accommodate each other’s needs ( United States v. AT&T )
The court dismissed the case, without reaching the executive privilege
claim, on the ground that judicial intervention in a dispute “concerning the respective powers of the Legislative and Executive Branches ... should be delayed until all possibilities for settlement have been exhausted...Judicial resolution of this
constitutional claim ... will never become necessary unless Administrator Gorsuch [Burford]
becomes a defendant in either a criminal contempt proceeding or other legal action taken by
Congress. ( United States v. House of Representatives)
**The Miers case where Bates agreed that she was compelled to submit to the Congressional subpoena but could exercise executive privilege for specific questions was very limited in its scope.
For it to be applicable here, the justification for the subpoena (legitimacy of the investigation) , as contested by the Administration ( Eggleston said the Oversight Committee's interest in the poltiical office "lacks any predicate of wrongdoing or misconduct.), would need to be established by a court. Bates trounced the idea of absolute immunity for advisers but did acknowledge qualified immunity as a legitimate potential claim.
senior presidential advisors are entitled to qualified immunity against damages actions. The qualified immunity inquiry,however, does not fit comfortably in the present context...Similarly, it might apply where Congress is not utilizing its investigation authority for a legitimate purpose but rather aims simply to harass or embarrass a subpoenaed witness.
Basically, its too early to tell how this situation is going to play out.
Here's the letter from Eggleston to Issa.