They began three days ago – initially in protest at falling living standards – and are the biggest show of dissent since huge pro-reform rallies in 2009.
Demonstrators have ignored a warning by Iran’s interior minister to avoid “illegal gatherings”.
Two demonstrators are reported dead in Dorud after sustaining gunshot wounds in a video posted on social media.
Videos filmed elsewhere in the country show protesters setting fire to police vehicles and there are reports of attacks on government buildings.
The current protests began in Mashhad on Thursday over living standards and rising food prices, and by Friday had spread to several major cities.
The Iranian authorities are blaming anti-revolutionaries and agents of foreign powers for the outbreak.
Overall, the numbers said to be taking part in demonstrations range from hundreds in some places to thousands in others – but demonstrations do not appear to be taking place on a massive scale.
Slogans have been chanted against both Mr Rouhani and Mr Khamenei, and clerical rule in general.
Demonstrators were reportedly heard on Friday yelling slogans like “The people are begging, the clerics act like God”.
There is also anger at Iran’s interventions abroad. In Mashhad, some chanted “not Gaza, not Lebanon, my life for Iran”, a reference to what protesters say is the administration’s focus on foreign rather than domestic issues.
Much of the information about what is occurring is emerging on social media, making it difficult to confirm anything.
In the town of Abhar in northern Iran, demonstrators have set fire to large banners bearing the picture of Iran’s Supreme Leader Ayatollah Ali Khamenei.
Meanwhile in Arak in central Iran, protesters have reportedly set fire to the local headquarters of the pro-government Basij militia.
In the capital Tehran, large numbers of protesters gathered at Azadi square, BBC Persian reports. A senior Revolutionary Guards’ figure in Tehran said the situation in the city was under control.
Demonstrators would be met with “the nation’s iron fist” if they continued, Brigadier-General Esmail Kowsari told student news agency ISNA.
In Mashhad, in the north-east, protesters burned police motorcycles in a confrontation caught on video. (Courtesy BBC)