Does the one who starts to pray Taraweeh have to complete it?

There is no doubt that Taraweeh is a Sunnah and is naafil;
it is the qiyaam of Ramadan. The same may be said of night prayers, Duha
(forenoon) prayer and the regular Sunnah prayers that are offered along
with the obligatory prayers. All of them are naafil, so he may do them if
he wants or not do them if he wants, but doing them is better.

If he starts to pray Taraweeh with the imam and wants to
leave before completing it, there is nothing wrong with that, but staying
with the imam until he finishes is better, and it will be recorded for him
as if he spent this night in prayer, because the Prophet (blessings and
peace of Allah be upon him) said: “Whoever prays qiyaam with the imam until
he finishes, Allah will record for him the qiyaam of one night.” So if he
stays with the imam until he completes it, he will have attained the virtue
of spending the whole night in prayer, but if he leaves after praying some
rak‘ahs, there is nothing wrong with that and no sin, because it is naafil.
End quote. 

Shaykh ‘Abd al-Azeez ibn Baaz (may Allah have mercy on him)