The provocative magazine, which suffered a deadly jihadist attack in 2015 after publishing cartoons of the Prophet Mohammed.
The provocative magazine, which suffered a deadly jihadist attack in 2015 after publishing cartoons of the Prophet Mohammed, depicted Ramadan with a huge erection in its edition last Wednesday, saying: "I am the sixth pillar of Islam."
The Swiss academic, an Oxford professor and conservative Islamic intellectual in France, has been accused of rape by two women after the Harvey Weinstein scandal unleashed a wave of sexual abuse accusations worldwide.
Ramadan, 55, has furiously denied the accusations as a "campaign of lies launched by my adversaries".
"Rape," reads the caption on Charlie Hebdo's cover. "The defence of Tariq Ramadan."
Laurent "Riss" Sourisseau, the magazine's editor, said the threats and hate mail had "never really stopped" after the January 2015 jihadist attack in which 12 people were gunned down at its offices.
"Sometimes there are peaks when we receive explicit death threats on social media -- this has been the case once again," he told Europe 1 radio.
"It's always difficult to know if these are serious threats or not, but as a principle, we take them seriously and press charges."
The shooting at Charlie Hebdo was claimed by Al-Qaeda, with the jihadists notably seeking to punish the staunchly atheist magazine for printing cartoons of the prophet Mohammed, forbidden in Islam.
The attack was the first in a wave of jihadist attacks in France over the past two years that have left more than 240 people dead.
Charlie Hebdo has continued to court controversy since the attack, notably with cartoons after the Barcelona attack and others that made light of an Italian earthquake that killed nearly 300 people.
The five pillars of Islam are the five basic practices obligatory for believers: faith, prayer, charity, fasting, and the pilgrimage to Mecca.
A minority of Sunni scholars considers jihad -- or holy struggle, which can have different meanings -- to be the sixth pillar of Islam.