Protests and counter-protests have been taking place for weeks in the valley and in the mainly Hindu region around the city of Jammu further south.
The demonstrations in the valley are some of the biggest since a separatist rebellion against Indian rule broke out nearly 20 years ago.
More than 20 people - Muslims and Hindus - have been killed and hundreds wounded in clashes with police since the unrest began in June.
Security forces opened fire on Tuesday on a number of protests by Muslims who defied the curfew.
The violence is spreading
The army fired to disperse a procession in the northern district of Bandipora, killing three people and injuring five. Three other people were killed in the Lasjan and Rainawari areas.
"We have imposed a curfew to ensure the peaceful burial of Sheikh Aziz," local police chief Kuldeep Khuda told reporters.
Shops, businesses, schools and colleges in the valley have all been closed, and other separatist leaders are under house arrest.
In the Jammu region, one person was killed and more than a dozen injured after Hindus and Muslims clashed in the town of Kishtwar. Houses, vehicles and other property were damaged before police opened fire to restore order.
No other violence has been reported from the Jammu region.
Sheikh Aziz was a prominent leader of the All Party Hurriyat Conference, an umbrella of separatist groups which opposes Indian rule.
Large crowds gathered at the main mosque in Srinagar to attend his funeral. Mourners shouted slogans against India including "We want freedom".
Police say they are trying to find out why security forces opened fire on protesters on Monday.
Police say several of their personnel were injured by stones thrown by those at the protest.
Sheikh Aziz was among thousands of protesters who marched on Monday towards the Line of Control (LoC), the de facto border with Pakistani-controlled Kashmir.
Police open fire on protesters and those killed on Monday are laid to rest
They were supporting fruit growers seeking to sell their produce. Muslims in the Kashmir valley complain that supplies of food and medicines are running short.
And they say crops of fruit which should have been sent to market are rotting away because Hindus are blockading a key road that links the valley with the rest of India.
The government, which denies the blockade, says lorries are moving between the two regions with security escorts.
Violent demonstrations began two months ago in the state when a decision to transfer a small area of land to the trust which runs a Hindu shrine provoked an angry Muslim reaction.
When the land transfer was abandoned, groups from the state's Hindu minority began furious protests of their own.
Our correspondent says the government was slow to realise how volatile the situation had become - and opposition parties were quick to try to use the situation to their own political advantage.
The result is that Kashmir is more divided along communal lines than it has been for years.
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