The protesters, part of the Ultima Generazione (Last Generation) group, stuck their hands to Botticelli's Primavera (Spring).
The Italian museum said no damage was done to the Renaissance painting.
In a video shared on social media on Friday, the group can be seen holding up a banner reading: "Ultima Generazione, no gas, no carbon."
The three protesters, who paid for tickets to the museum, were reportedly escorted off the premises by police.
"Is it possible to see a spring as beautiful as this today?" Ultima Generazione said in a statement.
"Fires, food crises and drought make it increasingly difficult. We decided to use art to sound an alarm call: we are heading towards a social and climate collapse."
In the video, people can be heard shouting: "Don't touch it," in English.
Primavera is on show in the gallery's Botticelli hall, along with The Birth of Venus.
This is not the first time climate activists have used art to draw attention to their protests.
A male visitor, disguised as an old lady, tried to smash the glass protecting the famous painting before smearing cream across its surface.
A video posted on social media showed a member of staff cleaning the pane while another attendant removes a wheelchair from in front of the masterpiece.
"Think of the Earth, people are destroying the Earth", the man, dressed in a wig, said in French in another video.