The Senate is the judge, pursuant to the Constitution of the "elections, returns and qualifications of its members". While some debate the question, the better interpretation appears to be that the Senate's powers with regard to appointments are coextensive with its powers with regard to elections. That is, the Senate has the same power to determine to seat or not to seat an appointed Senator as it does with respect to an elected Senator.
In Powell v. McCormack, the Supreme Court determined that the House's power to not seat a representative because of a failure to meet the qualifications for membership was limited to a failure to meet the enumerated Constitutional qualifications. However, Powell does not deal with the power to judge elections and returns.
It is reasonably clear that the Senate has the power under the Constitution to judge the bare fact of whether someone has been validly elected or appointed to the seat. That is, the Senate may judge if someone in fact was appointed or elected (e.g. received the most votes) and if the election or appointment was made by the correct authority (e.g. the Governor of Illinois). Beyond that, it appears that the Senate's power would extend to the question of whether the election or appointment was the result of fraud or bribery. See Barry v. United States ex Rel. Cunningham, 279 U.S. 597 (1929) at 615.
(Two interesting points about the Barry case: First, it seems clear that the Senate may delay seating the putative Senator until its investigation is completed. Second, the Supreme Court describes the Senate's power in this area as "beyond the authority of any other tribunal to review". Id at 613. However, the jurisprudence in this area has evolved significantly since 1929 and the degree to which the political question doctrine is respected has eroded significantly. Thus, it is likely that post-Powell the courts would find that the Senate's determination is reviewable.)
While the Senate has the power to refuse to seat a Senator selected as a result of bribery, there has been no credible allegation here that Burris was selected because he bribed the Governor of Illinois. Those who oppose seating Burris make instead the argument that the process of his selection was tainted because Blagojevich sought bribes from other Senate contenders.
Letting the Senate exclude a Senator because the process was tainted is problematic (to put it mildly) on a couple of levels. First, let us imagine the Blagojevich is not impeached. As impeachment is a difficult process, this is certainly possible. In that case, the implication would be that the Senate could refuse to seat the Senator from Illinois for two years.
Indeed, given Reid's statements it is clear that this is exactly the position of the Senate Democrats. This goes beyond saying a particular appointment was tainted to usurping an appointing authority's power. Leaving aside the question of whether it is desirable to enable the Senate to deprive a state of a Senate seat for two years, in what sense then, will the Senate be judging the "elections and returns" of Burris?
This situation becomes more analogous to Powell. In Powell, the House admitted that Powell met the Constitutional qualifications, but determined not to seat him because he was otherwise unqualified. Here, the Senate would be admitting that Burris was duly appointed (i.e there was no problem with the returns or elections of Burris), but that for other reasons, his election was invalid. The likely result is that the courts will find that Burris should be seated. (And the likely result of that is that the Senate Democrats will look bad at the end of the day.)
More broadly, what is it that would be considered a tainted process? In this case, there is nothing about the process of Burris' appointment itself that is problematic. Rather, it is other conduct of the appointing authority that is at issue. If that is relevant, we are, at the least, permitting the Senate to consider other corrupt conduct of a governor in considering whether to seat a Senator appointed by that governor. At the most we are raising the question of what might taint an election beyond illegal conduct in the election at issue. I would suggest that we are probably better off not going down this route and, absent at least some evidence that Burris was selected as a result of bribery, seating Burris.