Shenouda died on Saturday in Cairo at age 88 after suffering from what the official MENA news agency says was liver and lung disease.
Mourners gathered in the main Coptic cathedral in Cairo to pray and try to get a look at the pope's body. Many wept openly.
The grand sheik of Cairo's Al-Azhar, the pre-eminent theological institute of Sunni Islam, Sheikh Ahmed el-Tayeb said, "Egypt has lost one of its rare men at a sensitive moment when it most needs the wisest of the wise..."
Shenouda was born Nazeer Gayed, entering the priesthood and later a monastery as a young man. He was elected Coptic pope in 1971.
The late Egyptian President Anwar Sadat exiled Shenouda to a desert monastery in 1981 after he accused the government of failing to take action against Muslim extremists. Former President Hosni Mubarak ended his exile in 1985.
Shenouda led Copts through periods of tension with Egypt's Muslim majority. Islamic hardliners have carried out attacks against Christian churches, including bombing a Coptic church in Alexandria in 2011, killing 23 people.
U.S. President Barack Obama said Saturday he is saddened by Shenouda's death, calling him an "advocate for tolerance and religious dialogue."
Secretary of State Hillary Clinton says the United States sends its deepest condolences to the Egyptian people.
Many Copts are concerned that violence and persecution may continue after conservative Islamists won a large a majority of seats in recent parliamentary elections.