Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members
But the only qualifications listed in the Constitution are these in Section 3:
No Person shall be a Senator who shall not have attained to the Age of thirty Years, and been nine Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State for which he shall be chosen.
The election/appointment to Congress is the role of the state, not the Federal government. If the appointee meets the above qualifications, which Burris does, the Senate has no grounds on which to deny him a seat. While the Governor of IL has been indicted and charged with attempting to sell this seat, he has not been convicted or impeached, he denies the charges and is still legally the governor, so he has the power to fill the vacancy. There is a space on the appointment certificate for the signature of the state's Secretary of State, and that space is blank -- but regardless of Senate rules, there is no requirement for anyone but the governor to approve the appointment, so using this argument is specious.
Where they are right is morally. Anyone who would accept an appointment which was strongly suspected of being for sale does not have the moral standards one would want in a senator. Burris is a cog in the Chicago Democratic machine, which also does not speak well of him. But if the Senate were to kick out every member who has some nebulous moral failing, they would never field a quorum.