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 – in which a crying baby is heard at some points – 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.
Analysis – Inayatulhaq Yasini, BBC Pashto
It is the first time we have seen such differences among the group’s leadership.
Finding a unifying leader like Mullah Omar will be almost impossible for the Taliban, so a split is likely.
Mullah Mansour’s supporters have dispelled reports that military commander Qaum Zakir is opposing his election.
Zakir is an ex-Guantanamo prisoner, who has a base of support in Helmand province and will play a crucial role in the group’s future.
The side which manages to gains the support of military commanders will win the majority share of Mullah Omar’s movement.