A series of rows over Internet access have in recent months underscored the gulf between Iranian moderates, who seek fewer online restrictions, and conservatives who want more.
In Iran, Zuckerberg has been dubbed the "Zionist manager" of Facebook, on account of his Jewish heritage.
The court in Fars province has ordered the American tech tycoon to address violation of privacy claims over the reach of the Facebook-owned WhatsApp and Instagram services, ISNA news agency reported.
"Based on the judge's verdict, the Zionist manager of Facebook... should report to the prosecutor's office to defend himself and make compensation for damages," Rouhollah Momen-Nasab, a senior Iranian Internet security official, told ISNA.
"Following a complaint lodged by some of our fellow countrymen over the violation of their privacy and problems ensuing from WhatsApp and Instagram, the judiciary official has ordered a ban on these two software devices," he said.
However, a prosecutor in Shiraz, the provincial capital, denied Zuckerberg had been summoned, though claims against WhatsApp and Instagram seeking "the release of pictures and films," were being investigated.
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"There are complaints against these sites for Internet fraud and release of obscene photos and the plaintiffs have asked the judiciary to identify those who are involved," prosecutor Ali Alqasi told the official IRNA news agency.
Neither WhatsApp or Instagram have been blocked, he said.
Access to social networks, including Twitter and Facebook, are routinely filtered by Iranian authorities, as are other websites considered un-Islamic or detrimental to the regime.
President Hassan Rouhani, a self-declared moderate, has promised greater tolerance on social, cultural and media issues -- a vow that helped him defeat conservatives in last year's election.
But his fledgling push has been opposed by traditionalists and ultra-conservatives that hold sway in the establishment and key institutions.
Officials have voiced support for lifting the wider ban on social media, with some of them having Facebook or Twitter accounts.
Rouhani this month vetoed a plan to ban WhatsApp, preventing implementation of curbs sought by Iran's Committee for Determining Criminal Web Content.