Holding the Mushaf In the Taraweeh
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: http://alifta.com/Search/ResultDetails.aspx?languagename=en&lang=en&view=result&fatwaNum=&FatwaNumID=&ID=1898&searchScope=14&SearchScopeLevels1=&SearchScopeLevels2=&highLight=1&SearchType=exact&SearchMoesar=false&bookID=&LeftVal=0&RightVal=0&simple=&SearchCriteria=allwords&PagePath=&siteSection=1&searchkeyword=104111108100105110103#firstKeyWordFound
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