In practice, their work ends at midnight on 31 December.
Russia also refused to endorse the OSCE's budget for next year, meaning that the organization will have to limp along on emergency funding until the crisis is resolved.
OSCE rules require the consensus of all 55 member states for decisions to be approved.
OSCE spokesman Richard Murphy said border monitors will begin to leave their posts starting tomorrow and continue the process of withdrawal over the weekend. He said Georgia, the United States, the European Union, and others have pledged to continue to pressure Russia to reverse its veto.
"There was no consensus on extending the border-monitoring operation in Georgia," Murphy said. "So as of 31 December, the mandate of the border-monitoring operation will expire. Negotiations will continue in January on this and other unresolved issues, such as the 2005 budget."
The monitors are assigned and paid for by individual states -- not by the OSCE.
Murphy said the withdrawal of the monitors will be managed in such a way that they could rapidly return to their posts if a compromise is achieved.
Diplomats in Vienna told RFE/RL that Russia made it clear several days ago that it would refuse to extend the mandate of the monitoring mission. A compromise proposal that would have extended the mandate until 30 June failed.
The OSCE's Georgian border missions began in 1999 when Moscow charged that armed Chechen fighters were traveling back and forth across the border with Georgia. More than 30 nations contributed unarmed monitors to the mission, including Russia.
European Union representative Daan Everts said today that it is "too soon for funeral rites." He said the EU might be willing to supply its own force of monitors to work alongside Georgian monitors. Several delegates suggested that this could create an unfavorable situation for Russia because Moscow would have no control over such an effort.
Concerning the budget, Russia said it would approve it only if the OSCE accepted Moscow's proposal for a special conference on protecting energy supplies and also agreed to a discussion on changing the OSCE's system of monitoring elections. Moscow has criticized the OSCE's conduct of election monitoring in Belarus, Ukraine, and Uzbekistan.
Murphy said the Russian veto will curtail OSCE operations but not stop them. Regulations allow the OSCE to spend 25 percent of last year's budget in the first three months of the new year.
He recalled that Russia also vetoed the budget in 2002 in a dispute involving Latvia and Estonia. It lifted its veto five months later.
Slovenia, which is taking over the chairmanship of the OSCE, said the failure to approve a budget for next year threatens to damage the organization's international credibility.