It is permitted to recite Tehillim at night for a great need, or when there is a sick person or a calamity, Heaven forfend.
Sources and Explanations
In Chazon Ovadia (Avelut, vol. 1, p. 19), he ruled that it is permitted to recite from the book of Tehillim by way of supplication and request even before midnight for the sake of a sick person, and those reciting should have in mind that the merit of the study of Tehillim should protect the sick person, and that Hashem, may He be blessed, should send His word and heal him. In Be’er Moshe (vol. 4, siman 22), he was likewise lenient to recite for the sake of the sick person, and moreover there is a mitzvah in this recitation according to all opinions. He added that in his opinion, even the Chida, who was stringent in this matter because of the words of the Arizal, would agree that in a case of illness one should not be stringent at all.
Likewise, the Eshel Avraham (of Butchach, siman 306) was lenient to recite it for a sick person.
From the sefer “Shaf Veyativ – Laws of Saying Tehillim” by Rabbi Shay Tahan, Chapter 2: The Proper Times for Saying Tehillim, Halacha 14 (p. 24 in the print edition). Translated from the Hebrew original.
