The move re-exposed persistent tension between President Hassan Rouhani's government and lawmakers in Tehran, where hardliners routinely voice doubt about the merit of talking to the West.
One of Rouhani's vice presidents said the bill, which still has to be signed into law by Iran's Guardian Council, was unconstitutional and would not help its negotiators at "a sensitive point" in the talks.
The move, just one week before the deadline for an international agreement on the Islamic republic's disputed nuclear programme, lays down terms for MPs' accepting it.
However, critics of his nuclear policy, including members of the conservative-dominated legislature, say too many concessions have been made and, using the bill, they demanded a bigger say.
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The draft law says the government must "preserve the country's nuclear rights and achievements", a reference to retaining the ability to enrich uranium and keeping all nuclear facilities open.
Such demands have already been enshrined in an outline agreement struck on April 2 between Iran and the P5+1 powers -- Britain, China, France, Russia and the United States plus Germany.
The timing of removal has become important as members of the P5+1 have said it can only happen upon international verification that Iran has met requirements laid down under a deal.
Some 214 lawmakers out of 244 present supported the bill, with 10 against, six abstentions and the remainder not voting.
Bargaining over a final deal, due by June 30, intensified Monday when Britain and France reiterated that comprehensive inspections of Iran's nuclear facilities are essential.
Vice President Majid Ansari, who was prevented from addressing parliament, later told state media the terms of a nuclear deal were not a matter for lawmakers.
"The engagement of MPs on this issue is of no help to the nuclear team and it could create problems in the negotiation process," he said, citing an infringement of Article 176 of the constitution.