The paper on Monday announced it was pressing charges after a rise in threats and abuse following the publication of last Wednesday's front cover.
It carried a cartoon of Ramadan, sporting an enormous erection and declaring "I am the sixth pillar of Islam" under the banner "Rape: Tariq Ramadan's defence".
The Swiss academic, an Oxford professor and 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 dubbed the accusations a "campaign of lies launched by my adversaries".
Tariq Ramadan took leave of absence on Tuesday from the University of Oxford in the wake of the allegations.
"By mutual agreement, and with immediate effect, Tariq Ramadan, Professor of Contemporary Islamic Studies, has taken a leave of absence from the University of Oxford," the university said in a statement.
Ramadan's duties will be reassigned and he will not be present on campus, the university said.
"An agreed leave of absence implies no presumption or acceptance of guilt and allows Professor Ramadan to address the extremely serious allegations made against him, all of which he categorically denies," the university said.
Twelve people were killed when Al Qaeda-linked gunmen attacked Charlie Hebdo's offices in January 2015, opening a series of attacks that claimed over 200 lives.
The paper's boss Riss (real name Laurent Sourisseau), who also drew the Tariq Ramadan cartoon, told French radio that the hate messages "have never really stopped" but there are occasional rises.
"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 inquiry will look into "written death threats" and "public glorification of an act of terrorism," according to sources.