Authorities said 30 people, including some opposition leaders, were injured in unrest in the capital after senators approved the bill in a secretive vote.
Opposition leaders denounced the vote Friday as a "parliamentary coup," saying it could clear the way for a return to dictatorship in the landlocked South American nation of 6.8 million people.
Furious protesters broke into the congress late Friday, ransacking lawmakers' offices and starting fires after senators approved a proposal to allow the president to run for re-election.
Right-wing President Horacio Cartes is seeking to amend the constitution to enable himself to run for office again in 2018 after his current term ends.
The measure requires approval in the lower house, the Chamber of Deputies, which delayed a vote originally set for Saturday.
Riots, fires, arrests
Rodrigo Quintana, 25, leader of the opposition Liberal Party's youth branch, was shot and killed as police searched the party's premises in Asuncion, party leader Efrain Alegre said.
The interior ministry said in a statement that "the authorities are investigating the circumstances of the death, which is presumed to have occurred at the hands of a member of the National Police."
It added: "We are going to establish fully what happened, and those responsible will be brought to justice."
Police raided the party offices after activists took refuge there during a night of riots, Alegre said.
The injured included three lawmakers, according to firefighters and an opposition senator. Police said 211 people were arrested, some of them minors.
To chants of "Dictatorship never again!" hundreds of protesters clashed with riot police, who used mounted units and water cannon to disperse them.
Calm was restored around 0400 GMT Saturday. Large numbers of police remained on alert.
Cartes's allies in the upper house of the legislature passed the bill on Friday, sidestepping resistance from opponents. Th
e vote took place in Senate offices as the main assembly hall was occupied by senators from the Liberal Party, opposed to the reforms.
Opposition senator Luis Wagner said those injured included Senate speaker Roberto Acevedo, lawmaker Edgar Ortiz, who was hit in the mouth by a rubber bullet fired by police, and Liberal leader Alegre, who lost to Cartes in the 2013 presidential elections.
Acevedo has challenged the bill in the Supreme Court, arguing it is unconstitutional.
The measure was scheduled to be considered Saturday in the Chamber of Deputies, where the president has a majority.
But after the rioting, the president of the lower house, Hugo Velázquez, announced the vote was postponed, saying he was shocked by the violence.
"I hope that calm and harmony will return," Velázquez said in a televised message.
History of dictatorship
Paraguay has banned presidents from re-election since 1992 to avoid a return to dictatorships like that of General Alfredo Stroessner, who ruled from 1954 to 1989.
Removing the ban would also allow left-wing former president Fernando Lugo to run again. He held power from 2008 to 2012, when he was removed after an impeachment trial.
If the latest measure is approved by the two houses, it is expected to be put to a referendum within three months.
The opposition condemned the move as a "parliamentary coup" and called for resistance.
"It is a dictatorial plan by Horacio Cartes with the complicity of Ferdinand Lugo," said Senator Carlos Amarilla.
Cartes blamed the violence on "a group of Paraguayans embedded in politics and the media aimed at destroying democracy and political and economic stability."
"Democracy is not won or defended by violence," he said on his Twitter account.
"We must not allow barbarians to destroy the peace, tranquility and welfare of the people."