The 20-year-old defender, on loan from Norwich, was reduced to tears in Friday night's 5-1 defeat at Anfield after something was apparently shouted at him from the Kop.
Merseyside Police and Liverpool, who have only just finished dealing with the fall-out from Luis Suarez's eight-match ban for racially abusing Manchester United's Patrice Evra, are continuing their investigations and Furman believes the perpetrators should be severely punished.
"I hope the authorities take heavy action because it was a disgrace," he said.
"He was abused and I'll leave it to Tom and the police to investigate that. What happened should not happen in the game."
Furman declined to confirm the insult was of a racist nature but after a police statement on Friday night clarified no-one had been arrested for racially aggravated behaviour his former team-mate Rueben Hazell claimed it was after speaking to Adeyemi on Friday night.
The Shrewsbury defender wrote on Twitter: "After the game I went into the Oldham dressing room to catch up with my old mates and spoke to Tom Adeyemi directly about what the altercation with the fan was all about.
"He said he was racially abused by the fan. This is FACT.
"He also said that police want to speak to him and was asking 4 advise (sic) from his team-mates what to do.
"Every1 around him told him 2 go ahead and give his version of events to the police.
"It's time for this kind of behaviour to stop. This isnt acceptable in football or in society and its time 4 black players 2 show a stand like young Tom did.
"As a 20yr old I thought he managed the situation extremely well and it was sad that the incident reduce him 2 tears in what should of been the happiest day of his footballing career to date."
Officials and players from both clubs would not comment on whether the abuse was of a racist nature but Furman hinted as much.
"It is difficult for me to say anything. It has been highlighted over the last few months and it is an absolute disgrace," he told BBC Radio Manchester.
"Tom, the authorities and police will sort out the situation but a lot of credit goes to Tom for the way he handled the situation."
Liverpool, Oldham and Merseyside Police are all working together in an attempt to discover the circumstances surrounding events last night.
On Friday night the club issued a statement which said: "An incident occurred in the second half of the match which is now being investigated by both the club and the police.
"We will continue to work with the police to establish the details of what actually happened and will make a further statement in due course."
Merseyside Police have little to add to their overnight position, when a spokesman said: "Merseyside Police can confirm that no-one has been arrested on suspicion of racially aggravated behaviour during the FA Cup fixture at Anfield.
"However, the Force and Liverpool Football Club are investigating following an incident that occurred during the second half of the game.
"The aim of the investigation is to establish the full circumstances surrounding the incident to ascertain if action needs to be taken."
It is understood police spoke to Adeyemi after the match and there talks between Oldham and Liverpool after the game.
The League One club will not comment further on the matter until they have received an update from police but they are happy with the way Liverpool dealt with the matter immediately and the importance they placed on the player's welfare.
Adeyemi's parent club Norwich spoke to him on Friday night to offer their support and have asked Oldham to keep them informed of developments, while it is understood they have also received assurances from Liverpool the matter will be investigated thoroughly.
Norwich chief executive David McNally wrote on Twitter: "Will pass on your support for Tom. He is ok.
"LFC have assured us that the awful incident will be fully investigated."
Norwich manager Paul Lambert revealed he had spoken to Adeyemi on Friday night and was confident the matter would be dealt with properly by Liverpool and the authorities.
"I spoke to him last night after the game," said Lambert, whose side beat Burnley 4-1 at Carrow Road on Saturday afternoon to reach the fourth round.
"He is okay, which is pleasing, but I am sure Liverpool Football Club will do it properly, investigate what has happened.
"I can't comment on what actually happened, but Tom himself, which is the most important thing, is okay.
"What we said will remain private, but he is fine, which is the main thing for me.
"He is only a kid learning the game, one of the nicest lads you will ever meet, a really intelligent lad and he is okay.
"I am sure Liverpool Football Club and the good people up there will investigate it and will deal with it accordingly.
"The main concern for me was Tom was okay."
On Saturday night Liverpool issued a statement outlining the work which has been done since Friday night to identify the culprit.
"Liverpool Football Club continue to work closely with Merseyside Police to establish all the facts surrounding the incident that occurred during last night's game.
"We take this matter extremely seriously and have today provided the police with the evidence we currently have available to us.
"This includes CCTV coverage and still photography of what we believe to be the relevant part of the stadium, statements from matchday stewards in that area and full access to all the footage captured by the cameras filming the match for our in-house tv operation.
"In addition, we have examined records of the tickets purchased in that part of the ground to see if they provide any additional information and have passed that on to the police.
"Finally, we would also urge any of our supporters who witnessed this incident or saw the individual involved and who have not already done so, to contact either the club or the police."