Helsinki accused its neighbour of channelling migrants to the crossings in retaliation for it joining Nato.
Kremlin spokesman Dmitry Peskov said Finland was making a "big mistake" and destroying bilateral relations.
Around 300 asylum seekers have arrived in Finland this week, according to border guards.
The Finnish Border Guard said barriers would be put up from midnight on Friday at the Vaalimaa, Nuijamaa, Imatra and Niirala border posts in south-east Finland.
These points have seen a surge in illegal crossings by citizens of countries including Syria, Yemen and Iraq.
Mr Peskov said: "(One can) only express deep regret that the Finnish authorities have taken the path of destroying bilateral relations," Russian state news agency TASS reported.
"Russia has never in modern history threatened Finland, we had no reason for any confrontation. Now they have chosen this path.
"From our point of view, this is a big mistake."
Finland's prime minister Petteri Orpo accused Russia of deliberately helping people without the proper documents to get to the border.
While confirming the closure of the four border crossings, those at Salla and Vartius in Finland's far north would remain open for asylum applications, Mr Orpo said.
One hundred asylum seekers arrived in Finland on Friday, according to border guards.
Finnish authorities say the people coming to their country arrive legally in Russia before travelling to the border to pass into Finland - an EU member state - and claim asylum.
The prime minister has accused the Russian authorities of engineering the crossings.
"It is clear that these people are helped and they are also being escorted or transported to the border by border guards," Mr Orpo said on Tuesday.
In 2021, thousands of migrants flew to Russian ally Belarus before crossing into European Union member states Poland and Lithuania. At the time, EU accused Belarusian leader Alexander Lukashenko of seeking to destabilise the bloc by facilitating the passage of third-country citizens into the 27-member bloc.
Finland shares a 1,340km (833-mile) border with Russia, Europe's longest.
Finland has closed four of its border crossings with Russia to try to halt a surge in asylum seekers it says was instigated by Moscow.