His voice is beautiful but he recites Qunoot in Fajr every Friday; should I pray Taraweeh behind him?

Firstly: 

The scholars differed concerning the ruling on reciting Qunoot in Fajr prayer every day. Some of them are of the view that it is Sunnah, and others are of the view that it is a reprehensible innovation (bid’ah). 

The second view is the correct one. That has been discussed in the answer to question no. 20031.  

Thus it is clear that what this imam is doing is something innovated for which there is no basis.

So he should be advised to think again about what he is doing, and he should follow one of the two opinions mentioned above: either to recite Qunoot in Fajr every day or to forego reciting Qunoot, depending on what evidence he thinks is most valid. 

Secondly: 

With regard to praying Taraweeh behind him, there is no reason why you should not do so, because the fact that he recites Qunoot in Fajr has nothing to do with Taraweeh prayer. Moreover, it seems that he recites Qunoot in Fajr on Friday on the basis of a misinterpretation, thinking that there is some special virtue in that. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) was asked: 

What is the ruling on going to different mosques, seeking out an imam with a beautiful voice, because of the khushoo’ (humility) and proper focus on prayer which result from that? 

He replied: 

It seems – and Allaah knows best – that there is nothing wrong with that, if the aim is to seek help by means of that in developing proper humility and focus in prayer, and to find joy in prayer and find tranquillity in his heart, because not every voice brings joy. So if the aim in going to hear the voice of one imam or another is the desire to seek good and to perfect one’s focus in prayer, then there is nothing wrong with that; rather it is to be appreciated, and he will be rewarded according to his intention. A person may attain khushoo’ behind one imam and not behind another, because of the difference between the two recitations and the two prayers. So if the intention behind going to a distant mosque is to listen to the imam’s recitation because of his beautiful voice, and to benefit from them, and so that he will focus properly in his prayer, and not just because of whims and desires or for a change of scene, rather it is for the purpose of benefit and knowledge, and seeking to attain khushoo’ in prayer, then there is nothing wrong with that. It is proven in the saheeh hadeeth that the Prophet (peace and blessings of Allaah be upon him) said: “The people who will receive the greatest reward for prayer is those who walk the furthest [to the mosque], then those who walk the next furthest.” So if his intention is also to increase the number of steps he takes, that is also a good aim. End quote. 

Fataawa al-Shaykh Ibn Baaz (11/328, 329). 

And Allaah knows best.