Seeing that we 'avoid making requests' on Shabbos, may one say Tehillim privately on shabbos for sick, not life-pressing, individuals? Is the halacha different if saying the mishaberach after Tehillim or for a zechus like another person's shidduchim?
Halichos Shlomo 14:11:19 says that one may say Tehilim on Shabbos for one that is sick if it is done privately (BeYechidus).
Beshivilai Bais Harafua based on Rabbi Shlomo Zalman Aurbach 7:15 says that if the sick person requests that Tehilim should be said for him, it may even be done with a Minyan.
Thanks to www.shut-halacha.co.il for this answer.