The dispute over the aircraft, which was shot down over Georgia's breakaway Abkhazia province on April 20, has added to growing tension between Russia and Georgia.
Ties between the two countries have been badly strained since Georgia stepped up its efforts to join Nato and Russia increased its support for the separatist regions of Abkhazia and South Ossetia.
Georgia's foreign ministry said its demand for compensation was made in a protest note handed to Russia's ambassador to Georgia on Tuesday.
Mikhail Saakashvili, Georgia's president, has called on Russia to withdraw the peacekeeping reinforcements it recently sent to Abkhazia and to rescind a presidential order strengthening Moscow's ties with Abkhazia and South Ossetia.
Speaking at a government session on Tuesday, Saakashvili said: "Russia initiates military actions against the country where it pretends to be a peacekeeper."
The fighter plane in the video footage has a distinctive twin-finned tail and the report, compiled by the UN mission in Georgia, concluded it was a Russian-made MiG-29 or Su-27, neither of which Abkhazia possesses.
The report said radar tracking showed that after the drone was shot down, the fighter jet headed into Russian airspace.
Russia's military has denied that one of its planes downed the drone, while Abkhazian officials have claimed their own forces shot it down.
Russia's foreign ministry said that the videotape "prompts serious doubts" and that the radar data used by the UN mission differed from Russian data.
In a statement, it also said research used in the report was untrustworthy and that there were doubts about the type of plane involved and its flight pattern.
Lavrov, speaking in Denmark, demanded that Georgia hand over the tape.
He said: "Until we receive it, we will not be able to provide a definitive response."
Lavrov said that Georgia had refused repeated Russian requests for the tape.
Abkhazia and South Ossetia have been under separatist control since the end of fighting with Georgia in the 1990s.
They are seeking either independence or absorption into Russia.