FYI. If the Supreme Court took the case, It isn't clear that the Senate has the Constitutional authority to refuse to seat a senator who has been validly appointed under the Constitution.The tricky part here is that Article 1 of the Constitution stipulates that the Senate is the "Judge of Elections, Returns and Qualifications" of its own members. The Senate actually has fairly broad latitude on questions of "Elections and Returns", which is why it could intervene, say, in the Minnesota recount (as it has done in similar cases in the past). An appointment, however, is not an election, which means the only vehicle open to the Senate is challenging the appointee's qualifications, and the Powell v McCormack precedent stipulates that such a review would be limited to his age, residency or citizenship. What the Senate would have to do instead is actually expel the member they just admitted to the chamber, which requires a 2/3 majority and would be much stickier in terms of precedent -- the Senate has not expelled a member since the Civil War.
Art I Section 5 says that "Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members..."
In Powell v McCormack, the Court held that the House of Reps couldn't refuse to seat Adam Clayton Powell as long as he met the Constitution's qualifications for membership (age, residency, citizenship.).
I guess, theoretically, the Senate could seat the appointee and then expel him with a 2/3rds vote. The Court wouldn't interfere on Political Question grounds: the Constitution doesn't specify the standard for expulsion so it is properly at the discretion of the Senate.
This places even more pressure on the Illinois Legislature to impeach Blagojevich, who in all likelihood is too delusional and/or too stupid to resign his seat. Impeachments have been a rarity in Illinois, and the state's Constitution does not establish specific grounds for a conviction on impeachment proceedings (that is, there's nothing analogous to "treason, bribery, or other high crimes and misdemeanors" as stated in the US Constitution). But nobody stands anything to gain by defending Blagojevich and he'd seem like a longshot to survive an impeachment trial, even if the legislature is somewhat making up the precedents as it goes along.
EDIT: Short of impeaching Blagojevich, the Illinois Legislature could also rewrite its appointments law in conjunction with establishing a special elections procedure, removing the governor's power to make an appointment, as some states like Oregon have done. See Adam B's legal explainer for more. FURTHER UPDATE: Crain's Chicago Business suggests this is exactly what the state legislature will try and do.