It would be unconstitutional for the Senate to seat Burris

[Freddie Stubbs]

The 17th Amendment of The Constitution states:

The Senate of the United States shall be composed of two Senators from each State, elected by the people thereof, for six years; and each Senator shall have one vote. The electors in each State shall have the qualifications requisite for electors of the most numerous branch of the State legislatures.

When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.

This amendment shall not be so construed as to affect the election or term of any Senator chosen before it becomes valid as part of the Constitution.

http://www.usconstitution.net/const.html#Am17

Illinois law does permit the Governor to make appointments, but it also requires such appointments to be certified by the IL Secretary of State. So far, IL SoS Jesse White has refused to do so.

Burris' only recourse is to take White to court and have a judge force his to do his job, as required by IL law. Until then, it is out of the Senate's hands.