In the answer to question no. 159418 we stated that the number of takbeers in the funeral prayer is four, all of which are pillars or essential parts of the prayer.
Based on that, if the imam says only three takbeers by mistake, he should not be followed, and it is essential to alert him to that. If he then says the fourth takbeer, his prayer is valid, but if he does not do that, then the people praying behind him have to complete the fourth takbeer, then say the salaam.
Al-Bahooti (may Allah have mercy on him) said: If a worshipper omits one of the four takbeers deliberately, his prayer becomes invalid, because he has omitted an obligatory part deliberately; hence it becomes invalid, as in the case of all other prayers. If he omitted it by mistake, he should say it; this is similar to the case if he says the salaam in an obligatory prayer before completing it by mistake – he should (correct his mistake) if it is soon afterwards, and his prayer is still valid.
End quote from Daqaa’iq Ooli an-Nuha, 1/362
Al-Bukhaari narrated, in the chapter on saying takbeer in the funeral prayer: Humayd said: Anas (may Allah be pleased with him) led us in offering the (funeral) prayer, and he said takbeer three times, then he said the salaam. It was pointed out to him, so he turned to face the qiblah, then he said the fourth takbeer, then he said the salaam.
Al-Haafiz (may Allah have mercy on him) said: ‘Abd ar-Razzaaq narrated from Ma‘mar, from Qataadah, from Anas, that he said takbeer three times in the funeral prayer, then he ended it by mistake. They said: O Abu Hamzah, you only said takbeer three times. He said: Get back into your rows. So they did so, and he said the fourth takbeer.
End quote from Fath al-Baari, 3/202
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: If the imam says the takbeer three times (in the funeral prayer) then says the salaam, what should the one who is praying behind him do? He replied: He has to alert him to that.
Question: What if he is not alerted to it?
Shaykh: If he is not alerted to it, then the one who is praying behind him should say the takbeer, and thus the communal obligation will have been fulfilled.
End quote from ash-Sharh al-Kaafi.
And Allah knows best.