It is allowed for the Imam to hold the mushaf while in Taraweeh.The proof for that is the following narration:
وكانت عائشة يؤمها عبدها ذكوان من المصحف
Aisha (may Allah be pleased with her) used to be led in prayer by her servant Dhakwaan from a Mushaf
[Reported by Bukhari in Mu’allaq form. Reported with a connected chain by Ibn Abi Dawood in Kitab al-Masaahif (no. 792) and others. The checker of the book, al-Waa’idh, graded it Saheeh (pg. 657)]
While it is allowed, it is better for him to recite from what he has memorized even if he repeats it again and again.
عَنْ سَعِيدِ بْنِ الْمُسَيِّبِ «أَنَّهُ كَانَ يَكْرَهُ أَنْ يَقْرَأَ الرَّجُلُ فِي الْمُصْحَفِ فِي صَلَاتِهِ إِذَا كَانَ مَعَهُ مَا يَقُومُ بِهِ لَيْلَهُ، يُكَرِّرُهُ أَحَبُّ إِلَيَّ
On the authority of Sa’eed al-Musayyib rahimahullah who said: that he disliked that a man read from the Mushaf in the prayer if he has [memorized] what he can recite in the night prayer. Repeating [what he has already memorized] is more beloved to me”
Reported by Ibn Abi Dawood (no. 814) in Kitab al-Masaahif and others. The checker of the book, al-Waa’idh, graded it Saheeh Li Gharihi (pg. 663)]
This is because it is closer to the Sunnah of the Prophet (ﷺ) and decreases the movements needed when putting the Mushaf down and carrying it again.
As for the one behind the Imam, it is not allowed for him/her to hold the mushaf. The Scholars have ruled against it. The fatwa of Shaikh Bin Baz rahimahullah on this issue can be seen here:
Q: What is the ruling on the Ma’mum (person being led by an Imam in Prayer) carrying the Mus-haf (copy of the Qur’an) during Tarawih (special supererogatory night Prayer in Ramadan)?
A: I do not know of any basis for this. The preponderant opinion is that they should observe Khushu‘ (the heart being attuned to the act of worship) with tranquility and should not hold the Mus-haf; rather, they should place the right hand over the left as is the Sunnah (action following the teachings of the Prophet), with the right hand over the left wrist and forearm and placing them over the chest. This is the most preponderant opinion and the best. Holding the Mus-haf distracts from these acts of Sunnah. The heart and eyes may become busy following the words on the pages and this distracts them from listening to the Imam (the one who leads congregational Prayer). I think that not doing so is the Sunnah. The praying person should listen attentively and not hold the Mus-haf. If they have knowledge, they will be able to correct the Imam; otherwise, others will correct him. Should the Imam make a mistake and is not corrected, it does not matter, unless it is in Surah (Qur’anic chapter) Al-Fatihah (Opening Chapter of the Qur’an). It matters only in Al-Fatihah, because it is an essential pillar of the Salah (Prayer). However, if the Imam omits some Ayahs (Qur’anic verses) other than Al-Fatihah, this does not matter if there is no one praying behind him who can alert him to that. If someone holds the Mus-haf to correct the Imam when necessary, perhaps there is nothing wrong with that. But if each person holds a Mus-haf, this is contrary to the Sunnah.
The only exception is for the one who is correcting the Imam.
The proof is:
كان أنس يصلي وغلامه يمسك المصحف خلفه فإذا تعايا في آية فتح عليه
Anas ibn Malik radiallahu anhu used to pray and his servant used to hold the Mushaf behind him. If he got stuck in a verse, he used to correct him”
[Reported in Musannaf Ibn Abi Shaybah with a Saheeh chain]
And Allah Knows Best
Faisal Ibn Abdul Qaadir Ibn Hassan
Shaykh Muqbil rahimahullah advises those who hold the Mushaf in their hands during Taraweeh prayer in order to follow the Imam in his recitation that they should leave this action off and that it is something makrooh.
Ash-Shaykh Albany, rahimahullah, mentioned that carrying the Mushaf in the salaat leads to the person making a number of unnecessary movements in the salaat, for example putting the Mushaf in his pocket and taking it back out from his pocket, and as well opening and closing the Mushaf along with turning the pages when following the Imam. Likewise, the Shaykh rahimahullah mentioned that at times the person who is holding the Mushaf in his hands may have to place the Mushaf under his armpits, especially when he intends to raise his hands or to do ruku or sujood, and as a result, he displays his hands as if he is tied to something.
Using the Quran app while following the Imam in the salaat comes with more complications.
AbdulFattaah Bin Uthman