UEFA's control and disciplinary body met on Wednesday and announced the outcome today.
The banner displayed at the AC Milan Champions League match showed images of IRA hunger striker Sands and Scottish nationalist icon William Wallace, along with a set of lyrics that included the line: 'The terrorist or the dreamer?'
UEFA rules prohibit messages of a political and ideological nature being displayed in a stadium.
The Green Brigade insisted they were football-related as they were designed to protest against the Scottish Government's Offensive Behaviour at Football Act and Police Scotland's implementation of the legislation, which it claims has criminalised expressions of Irish politics.
But Celtic chief executive Peter Lawwell demanded supporters leave their political views at home after UEFA announced the action.
And manager Neil Lennon admitted his ''heart sank'' when he saw the display, arguing it was neither the time nor place for such a demonstration.
The club said in a statement today: "Clearly it is extremely disappointing that the club must pay another sizeable penalty following the actions of a small minority, particularly given the previous assurances which were made to the club and the widespread understanding of the likely outcome of such actions.
"Regrettably, due to previous charges being brought against the club, again following the actions of a small minority, the fines imposed by UEFA are increasing in scale.
"It goes without saying that such actions must stop now, before the club receives a competitive sanction or one which would affect our supporters attending European matches.
"We are Celtic supporters and we must now move on and look ahead to tomorrow's match against Hibernian. We are sure our fans will come together, unite with the club and support the team with the commitment and passion we are famous for, ensuring that Celtic Park once again provides a positive footballing occasion for all."
The club have already moved to separate the Green Brigade at Parkhead after a number of seats were vandalised, in the section containing Brigade members, and pyrotechnics were let off and thrown on the pitch when Celtic beat Motherwell at Fir Park last Friday night.
The club handed ''precautionary suspensions'' to 128 supporters preventing them from attending home and away matches, while 250 season ticket-holders housed in section 111 - the Green Brigade's corner of Celtic Park - will be moved to other parts of the ground or given refunds on season tickets.
Police made five arrests on Thursday over the Motherwell disturbances.
Celtic were fined 15,000 euros (£12,500) two years ago after supporters chanted pro-IRA slogans during a Europa League match against Rennes in France.
They were then handed a further 25,000 euros (£21,000) punishment after their followers unfurled a banner reading "F*** UEFA" and set off flares during a later match in the same competition at Italian side Udinese's Stadio Friuli home.
Another incident involving a flare, this time at Celtic Park for August's Champions League qualifier against Cliftonville, saw the Hoops receive another 5,000 euros (£4,200) penalty three months ago.