Opposition leaders have called for a boycott after a number of protesters died in clashes with security forces.
Under the current constitution, the president cannot seek re-election because he is over the age of 70 and has already served two terms.
President Sassou Nguesso first came to power in 1979.
He is now coming to the end of his second seven-year term. He won the last election 2009 with nearly 79% of the vote in a poll boycotted by half the opposition candidates.
Tens of thousands of people took part in a peaceful demonstration against the referendum in September.
Four people died on Tuesday, when security forces dispersed  further protests in the capital Brazzaville and the economic capital Pointe-Noire.
Protesters told the BBC the security forces used live ammunition and that army helicopters were deployed. Several activists and opposition leaders reported arrests and intimidation.
The outcome of Sunday's vote and the Congolese people's response to it will be watched closely in the region, as presidents in the neighbouring Democratic Republic of Congo and Rwanda are also expected to try to run for third terms in forthcoming elections, the BBC's Maud Jullien reports.
There have been months of unrest in Burundi, where President Pierre Nkurunziza was re-elected for a controversial third term in July.
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Among the changes being voted on are scrapping the presidential age and two-term limits.
Texting and internet services were cut and public meetings banned ahead of the referendum, residents said.
The opposition have been campaigning under the slogan "Sassoufit", a pun on the French expression for "that's enough" (ca suffit).
Congo-Brazzaville is holding a referendum on constitutional changes that would allow President Denis Sassou Nguesso to stand for a third term.