If the imam forgets some verses, how can he recite them so that he will have completed the Qur'aan?
The reader will not attain the reward for completing the Holy Qur'aan unless he has recited all of it and not omitted any part of it. If he omits a verse and does not recite it, then he has not completed the Holy Qur'aan.
The scholars even said that he should recite the Basmalah at the beginning of every soorah except Soorat Baraa’ah (al-Tawbah), and he should pay attention to that so that he will have completed the Qur'aan.
Al-Nawawi (may Allah have mercy on him) said:
He should always recite Bismillaah al-Rahmaan al-Raheem at the beginning of every soorah except Baraa’ah, as most of the scholars said that it is a verse, as it is written in the Mushaf and it is written at the beginning of all soorahs except Bara’aah. If he recites it then he has definitely completed the recitation, and if he omits the Basmalah he has omitted part of the Qur'aan according to the majority. End quote.
Al-Tibyaan, p. 61, 62
Based on that, it is essential for the reader to recite what he forgot of the verses, so that he may attain the reward for completing the Qur'aan.
The scholars (may Allah have mercy on them) mentioned two ways narrated from the Salaf (early generation) of reciting the verse or verses that the imam forgot during Taraweeh prayers:
1.He should put all these verses together and recite them all on the last night of Ramadan. This is what the imams used to do in Makkah in the past.
2.When he remembers the verse that he forgot, he should recite it [this applies if he is still reciting in the prayer] and then resume his recitation until he has finished. This was narrated from ‘Ali (may Allah be pleased with him).
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said, noting that the order in which it is prescribed to recite is waived in the case of forgetting:
… if he forgets some of the verses of the soorah when praying qiyaam in Ramadan (i.e., Taraaweeh), he should not repeat it [the entire soorah] or repeat what comes after it, but if he deliberately changes the order of the verses in the soorah and recites the later ones before the earlier ones, that is not permissible according to scholarly consensus.
Imam Ahmad (may Allah have mercy on him) was asked about the imam during the month of Ramadan omitting some verses from the soorah: did he think that the one who was praying behind him should recite them?
He said: Yes, he should do that. In Makkah they used to appoint a man to write down what the imam forgot of verses and so on, then on night of completion he would recite what he had omitted.
His companions -- such as Abu Muhammad (Ibn Qudaamah) -- said: Rather that is mustahabb in order to complete the Qur'aan and so as to attain a complete reward. The people of Makkah and Ahmad and his companions thought that if the one who had omitted some verses recited them later on when he remembered them, that was sufficient to be regarded as having completed the Qur’aan and to attain the reward, even if that meant the order of verses was not followed in this instance, because without reciting these verses, his recitation of the soorah could not be regarded as complete. This was narrated from ‘Ali: he forgot a verse from a soorah, then whilst he was still reciting he recited it, then resumed his recitation from where he had got to, and no one noticed that he had forgotten anything except those who were Haafiz (i.e., had memorised the entire Qur'aan).
Majmoo’ al-Fataawa, 21/410, 411.
And Allah knows best.