Mullah Mansour was named as the new leader on Thursday, after the death of former head Mullah Omar was confirmed.
But a Taliban spokesman told the BBC he had not been appointed “by all Taliban”, going against Sharia law.
The audio message said fighters should unite as “division in our ranks will only please our enemies”.
It also said that the Taliban would “continue our jihad until we bring an Islamic rule in the country”.
The 30-minute recording was released to journalists by Taliban spokesman Zabiullah Mujahid on Saturday.
Some Taliban figures have accused pro-Pakistani circles of imposing Mullah Mansour, who is known for his support for peace talks, on them.
But in the audio message, Mullah Mansour dismissed peace talks as “propaganda campaigns by the enemy”.
At least one Taliban faction would have preferred Mullah Omar to be succeeded by his son.
Another Taliban spokesman, Mullah Abdul Manan Niazi, said those who elected Mullah Mansour had not followed the rules.
“According to Islamic rule and principles, when a leader dies, a Shura (council) is called, then its leader is appointed,” he added.