According to the Quran previous nations were destroyed because of rejecting Allah, His signs, His Message and the Prophets (as) who were sent to them. For example, Allah says in Surah Fusilat:
But if they turn away, then say, “I have warned you of a thunderbolt like the thunderbolt [that struck] ‘Aad and Thamud.
[That occurred] when the messengers had come to them before them and after them, [saying], “Worship not except Allah.” They said, “If our Lord had willed, He would have sent down the angels, so indeed we, in that with which you have been sent, are disbelievers.” (Quran 41:13-14)
However, according to Shia Tafsir the previous nations were destroyed because of rejecting Wilayah of Ali.
We read in Shia tafsir Miratul Anwar:
ولا يخفي ان هلاك الامم كان بسبب ترك الولاية
It should be clear that the (previous) nations) were destroyed because of rejecting Wilayah [of Ali]
Tafsir Miratul Anwar, p. 188 (Scan in the end)
We read in the same book on page 161:
من سائر الآيات المشتملة على الصيحة مع ان الهلاك بالصيحة ايضا في الامم السالفة كان لعدم قبول الولاية
All the verses in which the ‘blast’ and in which the destruction of the previous nations with the ‘blast’ is mentioned, that was due to not accepting wilayah.
Some of such verses are:
[011:067] The (mighty) Blast overtook the wrong-doers, and they lay prostrate in their homes before the morning,-
[011:094] When Our decree issued, We saved Shu’aib and those who believed with him, by (special) mercy from Ourselves: But the (mighty) blast did seize the wrong- doers, and they lay prostrate in their homes by the morning,-
[015:073] But the (mighty) Blast overtook them before morning,
[023:041] Then the Blast overtook them with justice, and We made them as rubbish of dead leaves (floating on the stream of Time)! So away with the people who do wrong!
Posted with some extra information by 13S2010