Amar and Chafetz argue that if the Senate concludes that Burris was selected through a corrupt appointment process, it is perfectly permissible for it to exclude him, analogizing the process as one produced through bribery. This is an interesting argument. They seem to be right that if the appointment was made through bribery, the Senate could choose not to seat the appointee. The Senate is, after all, made the "the Judge of the Elections, Returns, and Qualifications of its own members."
In this area, one must distinguish between who gets to decide and what the Constitution requires of the decisionmaker. At least as a matter of original meaning,there is a strong argument that the Senate gets to decide whether the appointment here was legitimate, since it probably falls under the judging elections clause. It is possible that the Supreme Court would review the Senate's decision here -- whether that is contemplated by the original meaning or not -- although my guess is that the Court would at least confer substantial deference on the Senate.
A separate question is whether the Senate would be correct to refuse to sit Burris under the theory that Amar and Chafetz propose. Here, I have my doubts. It is true that Blagojevich's initial selections were influenced by bribery and corruption, but those picks were not made. There is no evidence that Blagojevisch's selection of Burris was tainted by those considerations. While one might doubt Blagojevich's character in general, that is probably not a sufficient reason to refuse to recognize his pick. Consider the following analogy. Amar and Chafetz write: "At the founding, Senators were elected by state legislatures. If the Senate believed that legislators in a given state had been bribed into voting for a particular candidate, the Senate could refuse to seat him." That is true, but if the same state legislature produced another candidate, without any evidence of bribery, the Senate should not be allowed to refuse to seat him.
Finally, one might wonder what would happen if the Senate excludes Burris and the case is taken to the courts. Would the deference that the courts are likely to confer on the Senate be enough to allow the exclusion? Hard to know. But whatever the result, the Senate would have been wrong to have excluded Burris.