Salaf us Saliheen, who used to offer salah or permitted it, by looking at the Mus'haf (Qur'an).

It is perfectly fine to offer voluntary (Taraweeh) prayer while reciting it from Mus'haf (Quran).

1. A servant named Zakwan used to lead Sayyidah Ayesha رضیﷲ عنها in prayer (during Ramzan)

[Sahih al Bukhari 1/140, Musannaf ibn Abi Shaybah #7216 sanad authentic]

Imam Ibn Hajar رحمہ اللہ said:

اسْتُدِلَّ بِهِ عَلى جَوازِ قِراءَةِ المُصَلِّي مِنَ المُصْحَفِ

This hadith is evidence of the permissibility of a praying person to recite the Quran by looking at the mushaf.

[Fath al Bari 1/285]

كانَ أنَسٌ يُصَلِّي وغُلامُهُ يُمْسِكُ المُصْحَفَ خَلْفَهُ، فَإذا تَعايا فِي آيَةٍ، فَتَحَ عَلَيْهِ

2. When Anas ibn Malik رضیﷲ عنه used to lead the prayer, his slave help him giving Luqmah holding the Qur'an.

[Musannaf ibn Abi Shaybah 7222 Sanad Hasan]

عائِشَةَ ابْنَةِ طَلْحَةَ «أنَّها كانَتْ تَأْمُرُ غُلامًا، أوْ إنْسانًا يَقْرَأُ فِي المُصْحَفِ يَؤُمُّها فِي رَمَضانَ.

3. Sayyidah Ayesha bint Talha رحمہا اللہ used to command a boy/slave to lead her in prayer whilst reading from the Qur'an in the month of Ramzan.

[Musannaf ibn Abi Shaybah #7218 sanad sahih]

Likewise senior students of Sahaba, Imam Hasan al Basri, Imam Mohammad ibn Sireen and Imam Ata ibn Rabah deem it permissible to recite from the actual mus'haf while offering Salah.

[Musannaf ibn Abi Shaybah under the chapter فِي الرَّجُلِ يَؤُمُّ القَوْمَ وهُوَ يَقْرَأُ فِي المُصْحَفِ " Regarding the man who leads the prayer while reading from the Mus'haf]

كانَ ابْنُ سِيرِينَ يُصَلِّي والمُصْحَفُ إلى جَنْبِهِ، فَإذا تَرَدَّدَ نَظَرَ فِيهِ

4. When Imam Mohammad Ibn Sireen lead the prayer he used to keep Qur'an close to him and whenever he gets confused in any verse, he corrects himself looking at the Qur'an.

[Musannaf ibn Abd ar Razzaq #3931 sanad sahih]

5. Imam Shihab Az Zuhri was asked: Is it allowed to lead the prayer while reading from the Qur'an? He replied:

لَمْ يَزَلِ النّاسُ مُنْذُ كانَ الإسْلامُ يَفْعَلُونَ ذَلِكَ»

Yes! people are doing this, since they accepted Islam.

[al Musahif la ibn Abi Dunya pg. 222, Sanad Sahih]

6. Yahya bin Saeed al Ansari رحمہ اللہ said:

لا أرى بِالقِراءَةِ مِنَ المُصْحَفِ فِي رَمَضانَ بَأْسًا، يُرِيدُ القُرْآنَ»

I do not see any problem in offering salah by looking at Mus'haf (Qur'an) in Ramzan.

[al Musahif la ibn Abi Dawood 780, Sanad Sahih]

7. Imam Hakam رحمہ اللہ said

فِي الرَّجُلِ يَؤُمُّ فِي رَمَضانَ يَقْرَأُ فِي المُصْحَفِ، رَخَّصَ فِيهِ

if a man leads the prayer reading from the Mus'haf (Qur'an) in Ramzan, then it is permitted.

[Musannaf ibn Abi Shaybah #7219, Sanad Sahih]

8. Imam Hasan al Basri رحمہ اللہ said

لا بَأْسَ أنْ يَؤُمَّ فِي المُصْحَفِ، إذا لَمْ يَجِدْ» يَعْنِي مَن يَقْرَأُ ظاهِرًا

There's no issue offering salah by looking at the Mus'haf (Qur'an) if there's no one to recite it aloud (helping the Imam by giving luqmah).

[Musannaf ibn Abi Shaybah #7222, Sanad Sahih]

9. Rabah bin abi Maroof reported: Imam Ata ibn Abi Rabah رحمہ اللہ said :

أنَّهُ كانَ لا يَرى بَأْسًا أنْ يَقْرَأ فِي المُصْحَفِ فِي الصَّلاةِ»

There's no issue who reads Qur'an by looking at it while offering Salah. 

[al Musahif pg. 459, Musannaf ibn Abi Shaybah #7221 sanad sahih]

Shaikh Zubair Ali Zai رحمہ اللہ said

There were some scholars like Hammad, Qatadah etc who disliked reading from the Qur'an while offering prayer. [ref #7230]

This narration's ruling is applicable only if a sahih ul aqeeda, Hafiz Imam leads the prayer by looking at the Mus'haf deliberately.

Secondary, what weight do these statements of scholars have in comparison of the Senior Tabain may allah have mercy upon them all.

The Giant of Deobandhiya, Anwar Shah Kashmiri and Masla'h Taraweeh.

Allamah Anwar Shah Kashmiri Deobandi writes the following about twenty rakah taraweeh

فصح عنه ثمان ركعات و أما عشرون فهو عنه عليه السالم بسند ضعيف وعلى ضعفه اتفاق

What is authentically proven from the Prophet ﷺ is the number of eight Raka'ah, and the report that states that the Prophet ﷺ offered twenty has been narrated with a weak chain which makes it “Da’eef’ (weak) and there is a consensus upon its weakness.

[Arf al Shadhi 1/29]

He also writes:

ولا مناص من تسليم أن تراويحه في كانت ثمانية ركعات ولم يثبت في رواية من الروايات أنه ي صلى التراويح والتهجد على حدة في رمضان

There's no escape from accepting it that Prophet ﷺ offered eight rakah of taraweeh, and it is not proven from any narration from the narrations that He ﷺ offered Taraweeh and Tahajjud separately in Ramzan.

[arf al Shadhi 2/208]

Anwar Shah Kashmiri writes regarding Taraweeh and Tahajjud being the same.

وانما يثبت تغاير النوعين اذا ثبت عن النبي صلی اللہ علیہ وسلم انا صلى التهجد مع اقامته باالتراويح.

Both of the prayers will be taken as different when it is proven that Allah's Messenger offered Taraweeh along with Tahajjud.

[Faiz ul baari, Sharah Sahih al Bukhari 4/23]

The people of knowledge discouraged a man to pray Tahajjud if he has offered Taraweeh.

Anwar Shah Kashmiri Writes:

ثم ان محمد بن نصر وضع عدة تراجم في قيام الليل وكتب ان بعض السلف ذهبوا الى منع التهجد لمن صلى التراويح

Furthermore, Mohammad bin Nasr compiled many chapters and penned that some salaf disallowed a man to pray Tahajjud who has already offered Taraweeh.

[Faiz ul bari, 4/24]

He Further said:

آخر الليل ، مع " يؤيده فعل عمر سه فإنه كان يصلي التراويح في بيته في أنه كان أمرهم أن يؤذوها بالجماعة في المسجد ، ومع ذلك لم يكن يدخل فيها ، وذلك لأنه كان يعلم أن عمل النبي ﷺ كان بأدائها في آخر الليل ، ثم نبههم عليه ، قال : إن الصلاة التي تقومون بها في أول الليل مفضولة عما لو كنتم تقيمونها في آخر الليل ، فجعل الصلاة واحدة.

Saydna Umar's رضیﷲ عنه approach also proves that Taraweeh and Tahajjud is same, because he used to offer prayer at the end of the night while urged people to commemorate and offer salah in the mosque, yet he didn't join them because he knew that Prophet ﷺ used to offer this prayer at the end of the night. Then, he made people aware about it and said the prayer (taraweeh) you offer at the early part of the night is less virtuous than offering prayers at the end part of the night. Hence, here, Saydna Umar has declared Taraweeh and Tahajjud similar and one prayer.

[Faiz ul baari, Sharah Sahih al Bukhari 4/24]

Anwar Shah Kashmiri refutes the fallacy of Deobandhiya who say Taraweeh and Tahajjud are different, He writes:

قال عامة العلماء : إن التراويح وصلاة الليل نوعان مختلفان ، والمختار عندي أنهما واحد وإن اختلفت صفتاهما ، كعدم المواظبة على التراويح ، وأدائها بالجماعة ، وأدائها في أول الليل تارة ، وإيصالها إلى الشحر أخرى ، بخلاف التهجد فإنه كان في آخر الليل ولم تكن فيه الجماعة ، وجعل اختلاف الصفات دليلا على اختلاف نوعيهما ليس بجيد عندي ، بل كانت تلك صلاة واحدة ، إذا تقدمت سُميت باسم التراويح ، وإذا تأخرت سميت باسم التهجد ، ولا بدع في تسميتها باسمين عند تغایر الوصفين ، فإنه لا حجر في التغاير الاسمي إذا اجتمعت عليه الأمة ، وإنما يثبت تغاير النوعين إذا ثبت عن النبي أنه صلى التهجد مع إقامته بالتراويح

In general, our hanafi scholars say: Taraweeh and Tahajjud are two different prayers but according to me, they're the same and one prayer even though they both have different attributes. Like, Taraweeh is offered with regularity, it is offered in congregation, it is offered in early part of the night and sometimes till Sahri but contrary to this Tahajjud is offered in last part of the night, there's no congregation in this prayer. having different attributes doesn't hold any credibility of being different prayers is not correct according to me. Although Taraweeh and Tahajjud are the same prayers. When it is offered early then it is called Taraweeh and when it's offered at the late night then it is called Tahajjud. There's nothing new to have two different names when the attributes are different. because there's no issue to have different names when the attributes are different. Reason being when ummah agrees then dispute in name has no harm in it. Hence, it will be proven two different prayers when it is proven from Prophet ﷺ that he offered Taraweeh along with Tahajjud.

[Faiz ul Bari, Sharah Sahih al Bukhari 2/567]