Déby, 68 – who had been in power for three decades – died after being shot as he battled rebels on the frontline.
The rebels have also objected the move, saying: “Chad is not a monarchy.”
Mahamat Idriss Déby Itno, also known as “General Kaka”, was in charge of the presidential guard and is to lead the country for 18 months until elections.
Experts have told the BBC and other broadcasters that the move is unconstitutional, and that the speaker of parliament should take over when a sitting president dies before organising elections.
Déby’s death was announced on state TV on Tuesday – a day after provisional election results projected he would win a sixth term in office at the helm of the oil-rich country.
The government and parliament have been dissolved. A curfew has also been imposed and the borders have been shut.
There are fears that the death could trigger political instability in the vast semi-arid country where the opposition is weak and divided.
media captionChad’s President Idriss Déby was one of Africa’s longest-serving leaders.
Correspondents say few people are on the streets of the capital, N’Djamena, even after the end of the overnight curfew.
Tanks and heavy artillery are still deployed around the presidency, with rebel group Fact (the Front for Change and Concord in Chad) saying it will continue its advance south.
Neighbouring African leaders have expressed shock and dismay at Déby’s death.
There is also concern about the negative impact on the international fight against Boko Haram militants and other jihadist groups linked to the Islamic State group and al-Qaeda in West Africa.
Chadian forces are considered to be the most effective of the Western-backed “G5” countries fighting Islamist militants in the Sahel region.
Nigeria has called for dialogue between all parties, including the rebels, to prevent the country from sliding into chaos.
In a statement, the French presidency described Mr Déby as a “brave friend” and said it affirmed its attachment to the stability of Chad. Over the years, France has deployed troops and fighter jets to push back Déby’s opponents.
Déby was an army officer by training and a former head of the military who came to power in 1990 through an armed uprising – and faced numerous challenges and many coup plots in the years that followed.
Observers say it is no surprise to hear he died on the frontline as he would often take charge during military confrontations.
A state funeral is due to take place on Friday.
Five things about Chad:
1) It is named after Lake Chad. This is the second-largest lake in Africa, but has shrunk by 90% since the 1960s. Its basin covers parts of Nigeria, Niger, Chad and Cameroon, and is a water source for between 20 million and 30 million people.
2) The Sahara Desert roughly covers one third of the country. Much of the north is desert and home to a mere 1% of Chad’s population. The south has large expanses of wooded savannahs and woodlands.
3) Remains of a seven-million-year-old human-like creature – or hominid – known as “Toumai” were unearthed in 2001. Its discoverers argued that Toumai was the oldest hominid known to science.
4) Chad became an oil-producing nation in 2003, with the completion of a $4bn (£2.87bn) pipeline linking its oilfields to terminals on the Atlantic coast. The industry has been plagued by allegations of corruption.
5) Agriculture is the main source of livelihood for most people – cotton is grown in the south, and exported to Europe and the US.