Firstly, some smart folks arguing that the Senate probably does have the right to exclude Burris:
Akhil Reed Amar and Josh Chafetz, Slate
Lyle Dennison, SCOTUSBlog
Jack Balkin, Balkinization
And here are some other smart folks arguing that no, the Senate probably does not have such authority:
Scott Lemieux, Lawyers, Guns and Money
Brian Kalt, Concurring Opinions
Eugene Volokh, Volkoh Conspiracy
The crux of the argument seems not to be, as I questioned originally, whether Burris' appointment constitutes an "election" as specified by Article 1, Section 5 of the Constitution. The Senate certainly has had no trouble challenging appointments before, although it hasn't done so (at least as far not as I can tell) since the Powell v McCormack decision in 1969. (Amar and Chafetz also argue that -- notwithstanding the question about what constitutes an "election", the appointment of Burris qualifies as a "return", which the Senate also has the power to judge under Article 1, Section 5). Rather the question in light of the Powell decision, is to what if any extent the Senate is restricted in its power to serve as the judge of such an appointment.
I do think that -- and, in case this isn't clear to anyone, this is just a layman's reading of the decision -- people may be underestimating the intended scope of Powell, which seems to have been trying to cast a fairly wide net. In particular, the opinion holds that:
Unquestionably, Congress has an interest in preserving its institutional integrity, but, in most cases, that interest can be sufficiently safeguarded by the exercise of its power to punish its members for disorderly behavior and, in extreme cases, to expel a member with the concurrence of two-thirds. In short, both the intention of the Framers, to the extent it can be determined, and an examination of the basic principles of our democratic system persuade us that the Constitution does not vest in the Congress a discretionary power to deny membership by a majority vote.Emphasis mine. The key word in that paragraph is probably "discretionary". One can imagine two extremes here. In one case, imagine that the governor of Arizona, a state which requires that an appointed senator be from the same party as the vacating one, attempted to nominate a candidate from the other party instead, in apparent contravention to the state law. Or, to draw a more plausible scenario, suppose that the appointee switched parties on the day before her appointment to circumvent that law. Clearly, it seems, the Senate would have the authority to strike this down.
At the other extreme, imagine that in Indiana, which has no such law, Evan Bayh decided to resign from the Senate, Governor Mitch Daniels decided to appoint a Republican to fill his seat, and that the Democratic majority in the Senate didn't like this and so voted to exclude Daniels' otherwise-qualified appointee merely because they didn't want another Republican in their chamber. I would hope that the Senate would be precluded from doing something like that.
What the Powell decision seems to hold is that, if the Senate wants to exercise its discretionary power to prevent someone from serving in the Senate, it has another mechanism to do that, which is expulsion. So the decision to exclude a member must in some sense be non-discretionary.
In the case of a question about a member's qualifications to serve in the Senate, there is a pretty clear bright line between a discretionary decision and a non-discretionary one, which is that the Constitution specifically establishes the qualifications for serving in the Senate, such as being of sufficient age and being a U.S. Citizen. But can such a bright line be established between a discretionary and non-discretionary action in the case of the Senate's right to judge a gubernatorial appointment?
I don't know. What the Burris folks will probably argue, if the case gets that far, is that the Senate's power would be limited to judging whether such an appointment was made in contravention of either state or federal laws. That is, a non-discretionary judgment is one involving the enforcement or interpretation of such laws, and in order to be allowed to render such a judgment, the Senate needs to put a legal question before itself.
That the Burris appointment appears to be legal under Illinois law, that the state has alternate mechanisms to preclude Blagojevich from making such an appointment (either removing him from office or requiring that the vacancy be filled by special election), and that neither Blagojevich nor Burris have yet been convicted of any wrongdoing, might tend tip the scales in favor of this being precisely the kind of discretionary decision that the Court was seeking to prevent in Powell.
If, on the other hand, Blagojevich were impeached before Burris had been sworn into the Senate, perhaps the Senate might then have the right to judge whether Burris' appointment had been nullified by that impeachment, and to exclude him if it so chose. Perhaps the Senate also has the right to judge whether Blagojevich's appointment is invalidated by the failure of Illinois' Secretary of State to certify it. These are fairly specific legal questions.
Nor am I suggesting that the Senate has no right to judge the Burris appointment on its face. But the "non-discretionary" component of that might be to judge whether the appointment was legal or not -- not merely to judge whether it was in poor taste.
At the very least, the people arguing for a narrower interpretation of Powell would seem to have to reconcile the way in which the Court distinguished exclusion from expulsion in that opinion. Amar and Chafetz argue that the distinction is merely that exclusion concerns a potential member who has not yet been seated, whereas expulsion concerns someone who already sits in the chamber. That is a perfectly commonsensical interpretation -- but it is not the one that the Court appears to have made in Powell.