A message shared on WhatsApp by an embassy spokesman said the consular section would be closed as of Monday "until further intimation".
The spokesman told AFP that the consular section typically processes about 1,500 visa applications a day.
In Islamabad, foreign ministry said the Afghan charge d'affaires had been summoned to "convey serious concerns over the safety and security of the diplomatic personnel of the embassy of Pakistan, Kabul, and its sub-missions."
Protests outside the Pakistani embassy are common, sometimes triggered by people unhappy about visa wait times or security while standing in long queues.
Tensions have soured further in recent days amid clashes along the border in Afghanistan's eastern province of Kunar.