It states in the ‘Sefer Minhagim’ “it is known that at night from after the stars come out until after ‘Chatzot’, (Halachic midnight,) Tehillim are not said, but the daily shiurim of Chumash and Tanya are said.”
In the Rebbe’s letters it is explained that when you learn Chumash with the commentary of Rashi, it is considered to be learning the Oral Torah.
We are not stringent about not saying Tehillim after ‘Chatzot’, as are many Kabbalists, since King David himself said “At midnight I will rise to thank you.” The problem is only until Chatzot, because the first half of the night is a time of ‘Gvurot’, (‘Severity’), while the second half of the night is a time of ‘Chassadim’, (‘Loving kindness’).
It is permissible to say Tehillim for a sick person, as this is not learning the Tanach, but praying. It is also permissible during the High Holidays at night, or when guarding the body of the deceased, or similar cases.