Du’a after Every Two Rak’ahs of Tarawih: Allowed?
Question

What is the ruling on saying du’a between one taslim and the next in tarawih prayer by saying “Subhanaka Rabbana wa bi hamdika Allahumma ighfir lana (Glory and praise be to You our Lord; O Allah, forgive us)”?
Praise be to Allah.Du’a after every two rak’ahs of tarawih
There is no basis in the Sunnah for saying du’a between one salam and the other during Tarawih prayer by saying “Subhanaka Rabbana wa bi hamdika Allahumma ighfir lana (Glory and praise be to You our Lord; O Allah, forgive us)”, and neither the imam or those praying behind him are obliged to do that, because worship is governed by the principle that nothing is permitted except that which was prescribed by the Messenger of Allah (peace and blessings of Allah be upon him). 
Constantly doing a particular act of worship at a particular time without any evidence for that comes under the heading of bid’ah or innovation. 
The Prophet (peace and blessings of Allah be upon him) said: “Whoever does an action that is not part of this matter of ours (Islam) will have it rejected.” (Agreed upon)
This great dhikr - “Subhanaka Rabbana wa bi hamdika Allahumma ighfir li (Glory and praise be to You our Lord; O Allah, forgive me)” – is something that the Prophet (peace and blessings of Allah be upon him) used to say when bowing and prostrating. 
Al-Bukhari (784) and Muslim (484) narrated that ‘Aishah (may Allah be pleased with her) said: The Prophet (peace and blessings of Allah be upon him) used to say when bowing and prostrating, “Subhanaka Rabbana wa bi hamdika Allahumma ighfir li (Glory and praise be to You our Lord; O Allah, forgive me )” following the command given by the Quran. 
What is meant by following the command given by the Quran is that he would do what was enjoined upon him in the Quran, where Allah says (interpretation of the meaning): 
“When there comes the Help of Allah (to you, O Muhammad against your enemies) and the Conquest (of Makkah). And you see that the people enter Allah’s religion (Islam) in crowds. So glorify the Praises of your Lord, and ask His forgiveness. Verily, He is the One Who accepts repentance and Who forgives.” [al-Nasr 110:1-3]
See how the Sunnah has been abandoned in this posture of the prayer, and how the people adhere to innovations! 
Other Innovations during Tarawih
Other innovations which are done during Tarawih include:  

When people say: “Salat al-qiyam, athabakum Allah ((It is) time for qiyam prayer, may Allah reward you)”.
When they say between each two rak’ahs, “Allahumma salli wa sallim ‘ala sayyidina Muhammad (O Allah, send blessings and peace upon our master Muhammad) in a loud voice.
When people recite Surah al-Ikhlas and al-Mu’awwidhatayn between each two rak’ahs, or the imam says “Subhan Allah (Glory be to Allah)” and the congregation behind him responds by saying, “Subhan Allah wa bi hamdihi, subhan Allah il-‘Azim (Glory and praise be to Allah, Glory be to Allah the Almighty).” 

None of these things were narrated from the Prophet (peace and blessings of Allah be upon him), hence the Standing Committee issued a fatwa to the effect that they are innovations and bid’ahs. (Fatawa al-Lajnah al-Daimah, 7/207-210)
For more, please see these answers: 38025 , 66558 , and 37753 . 
And Allah knows best.