All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Weeping out of fear of Allaah is a recommended act; there are many religious texts in the Quran and the Sunnah to this effect. For instance, ‘Abdullaah ibn Ash-Shikhkheer said, "I saw the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, perform the prayer, and his chest was wheezing like a boiling pot." [Al-Bukhari]
There are many religious texts to this effect. It cannot be said that crying out of fear of Allaah in the prayer is a religious innovation in and of itself. On the contrary, it is a good deed, as we mentioned earlier.
However, it is better to confine oneself to weeping in the prayer when praying alone. When one is in a congregational prayer or seen by people, he should try to withhold his tears to the best of his ability in order to be cautious of showing off. If he cannot withhold his tears, then there is no harm in that and he bears no sin for it.
Shaykh Ibn ‘Uthaymeen was asked the following question: "I often weep aloud in most of my prayers; does this invalidate my prayer?" He replied, "If weeping in the prayer is due to the fear of Allaah, remembering the matters of the Hereafter, or reciting the verses of the Quran that mention warnings and threats (to the sinners), then it does not invalidate the prayer."
For more benefit, please refer to fatwa 85706.
Allaah knows best.