Mr Burris (L) ran unsuccessfully for the US Senate in 1984
Barack Obama says he agrees with Senate Democrats that they should not accept the man chosen by Illinois Governor Rod Blagojevich to replace him as senator.
Mr Blagojevich is the subject of a criminal inquiry and has been charged with attempting to "sell" Mr Obama's now-vacant seat to the highest bidder.
The governor defied pressure and picked Roland Burris, the state's former attorney general, to fill the position.
The controversy is set to hang over the Senate when it convenes on 6 January.
Democratic senators have vowed to veto anyone appointed by Mr Blagojevich.
The president-elect said he agreed the Senate "cannot accept" a new senator chosen by Mr Blagojevich, adding that Mr Blagojevich himself should resign.
Mr Blagojevich, himself a Democrat, denies wrongdoing and has rejected calls for his resignation.
'Fine public servant'
Mr Burris, 71, became the first African-American to be elected to statewide office in Illinois when he won the 1978 election to be state comptroller.
He was state attorney general from 1991 to 1995 and made unsuccessful bids for the US Senate and the Illinois governorship, including in 2002, when he lost in the Democratic primary to Mr Blagojevich.
"Roland Burris is a good man and a fine public servant," said Mr Obama.
"But the Senate Democrats made it clear weeks ago that they cannot accept an appointment made by a governor who is accused of selling this very Senate seat. I agree with their decision."
Appearing with Mr Burris to announce his choice, Mr Blagojevich said: "Please don't allow the allegations against me to taint a good and honest man."
But Harry Reid, the leader of the Democrats in the Senate was not impressed.
"It is truly regrettable that... Governor Blagojevich would take the imprudent step of appointing someone to the United States Senate who would serve under a shadow and be plagued by questions of impropriety," said Mr Reid.
"Anyone appointed by Governor Blagojevich cannot be an effective representative of the people of Illinois and ... will not be seated by the Democratic Caucus."
However comments by Illinois Representative Bobby Rush indicated that charges of racism may hang over the Senate if it does not allow Mr Burris to take his seat.
"There are no African-Americans in the Senate. And I don't think anyone, any US senator who's sitting in the Senate right now, wants to go on record to deny one African-American from being seated in the US Senate," Mr Rush said.
The Constitution gives the Senate wide powers to determine who can be seated in the chamber, but the authority is not absolute.
In 1969, the Supreme Court ruled that the House of Representatives had acted unconstitutionally when it excluded a congressman, Adam Clayton Powell, who had been accused of financial impropriety.
Adding to the complications, the Illinois Secretary of State, Jesse White, has said he will refuse to certify the paperwork Mr Blagojevich must present to the Senate regarding Mr Burris's appointment.
An internal review conducted by the Obama team concluded last week that neither the president-elect, nor his staff, had had any "inappropriate discussions" with Mr Blagojevich about who should fill the seat.
Mr Blagojevich has vowed to "fight the false accusations" made by what he has termed a "political lynch mob".
The Illinois state legislature has formed a committee to investigate the possibility of impeaching Mr Blagojevich.
CHARGES AGAINST BLAGOJEVICH
Federal agents say Mr Blagojevich:
Tried to obtain campaign contributions in exchange for official actions
Tried to use state funds for the private purpose of inducing the Tribune Company to fire Chicago Tribune editorial board members critical of him
Tried to obtain personal financial benefits for himself in return for his appointment of a US senator
There have been calls from many politicians, including Mr Obama, for the governor to step down.
He was charged on 9 December with a number of offences including soliciting a bribe.
The charges relate to a variety of corruption schemes in which the governor was allegedly involved, including so-called "pay to play" deals - the doling out of jobs, contracts and appointments in return for campaign contributions.
On the day of his arrest, investigators released transcripts of conversations between Mr Blagojevich and others intercepted by court-authorised wiretaps.
In the conversations, the Democratic governor allegedly discussed offering Mr Obama's Senate seat in return for a well-paid position at a non-profit organisation or a group affiliated with trades unions, according to the affidavit.