Iran protesters rally again despite warning of crackdown

* Protesters chant: "Reza Shah, bless your soul", breaking taboo

* Rouhani says Iranians have right to protest

* Rouhani also warns against violation of public order

* White House says that "voices deserve to be heard" (Adds White House statement, paragraphs 15-16)

By Bozorgmehr Sharafedin

LONDON, Dec 31 (Reuters) - Anti-government protesters demonstrated in Iran on Sunday in defiance of a warning by authorities of a crackdown, extending for a fourth day one of the most audacious challenges to the clerical leadership since pro-reform unrest in 2009.

Giving his first public reaction to the protests, President Hassan Rouhani appealed for calm, saying Iranians had the right to protest and criticise the authorities.

But he warned, according to official media: "The government will show no tolerance for those who damage public properties, violate public order and create unrest in the society."

Tens of thousands of people have protested across the country since Thursday against the Islamic Republic's government and clerical elite.

Police in the centre of Tehran fired water cannons to try to disperse demonstrators, according to pictures on social media.

Demonstrations turned violent in Shahin Shahr in central Iran. Videos showed protesters attacking the police, turning over a car and setting it on fire. Reuters could not immediately verify the authenticity of the footage.

There were also reports of demonstrations in the western cities of Sanandaj and Kermanshah as well as Chabahar in the southeast and Ilam and Izeh in the southwest.

Demonstrators initially vented their anger over economic hardships and alleged corruption, but the protests took on a rare political dimension, with a growing number of people calling on Supreme Leader Ayatollah Ali Khamenei to step down.

Iranian security forces appear to have shown restraint to avoid an escalation of the crisis. Two people have been killed and hundreds arrested.

The protests were the biggest since unrest in 2009 that followed the disputed re-election of then-President Mahmoud Ahmadinejad.

Videos showed people in central Tehran chanting: "Down with the dictator!" in an apparent reference to Khamenei.

Protesters in Khorramabad in western Iran shouted: "Khamenei, shame on you, leave the country alone!"

The government said it would temporarily restrict access to the Telegram messaging app and Instagram, owned by Facebook Inc , state television said. There were also reports that mobile access to the internet was being blocked in some areas.

"Iran, the Number One State of Sponsored Terror with numerous violations of Human Rights occurring on an hourly basis, has now closed down the Internet so that peaceful demonstrators cannot communicate. Not good!" U.S. President Donald Trump tweeted on Sunday.