The announcement came as Washington stepped up its rhetoric against Tehran, accusing it of planning "imminent" attacks and deploying an aircraft carrier strike group with several nuclear-capable B-52 bombers to the region.
Iran said it was responding to the sweeping unilateral sanctions that Washington has reimposed since it quit the agreement a year ago, which have dealt a severe blow to the Iranian economy.
It said it would stop implementing some of the restrictions it had agreed with immediate effect.
Tehran said it would abandon more if the remaining parties to the agreement -- Britain, China, France, Germany and Russia -- failed to start delivering on their commitments to sanctions relief within 60 days.
President Hassan Rouhani underlined that the ultimatum was intended to rescue the nuclear deal from his US counterpart Donald Trump who has repeatedly called for it to be scrapped since he pulled out on 8 May 2018.
US Secretary of State Mike Pompeo described Tehran's announcement as "intentionally ambiguous".
Robert Kelley, a former UN nuclear inspector now with the Stockholm International Peace Research Institute, said the commitments Iran was dropping had no bearing on its ability to develop an atomic bomb.
He said Iran was simply seeking to "save face" after "striking a deal which was not respected by the other side."
US not world 'sheriff'
Under the landmark deal agreed by Trump's predecessor Barack Obama, the parties to the agreement were supposed to lift nuclear-related sanctions on Iran in return for it reining in its activities to ease fears it was seeking the capability to produce an atomic bomb.
But the promised sanctions relief has failed to materialise as European and Asian banks and oil companies have moved swiftly to abide by the renewed US sanctions for fear of financial or commercial repercussions.
Rouhani slammed European countries for seeing the US as the world's "sheriff" and said this keeps them from making "firm decisions for their own national interests".
The three European parties to the deal -- Britain, France and Germany -- tried to save the accord with a trade mechanism meant to bypass reimposed US sanctions, but their attempt was dismissed by Iran's supreme leader Ayatollah Ali Khamenei as a "bitter joke".
Foreign Minister Mohammad Javad Zarif, who was in Moscow on an official visit, accused European governments of not fulfilling their obligations under the nuclear deal.
"Our friends in Russia and China maintained very good relations with us in this year but the rest of the... participants did not meet any of their obligations," Zarif said.
Iran's Supreme National Security Council said it no longer considered itself bound by the agreed restrictions on stocks of enriched uranium and heavy water.
It said after 60 days, it would also stop abiding by restrictions on the level to which Iran can enrich uranium and modifications to its Arak heavy water reactor that were designed to prevent the production of plutonium.
Uranium enriched to much higher levels than Iran's current stocks can be used as the fissile core of a nuclear weapon, while heavy water is a source of plutonium which can be used as an alternative way to produce a warhead.
Small 'window for diplomacy'
The council called for swift action by the remaining parties to the deal, warning time was running out.
On the eve of the Iranian announcement, Pompeo made an unannounced visit to Baghdad, an ally of both Washington and Tehran.
Pompeo said he made the trip because Iranian forces are "escalating their activity" and said the threat of attacks was "very specific", without giving further detail.
Israel's Prime Minister Benjamin Netanyahu, a fierce opponent of the nuclear deal, seized on the Iranian announcement as evidence it was pressing ahead with its nuclear programme.
"We shall not allow Iran to obtain a nuclear weapon," he said.