Hummingbird –
From what I’ve read (I’m sorry I don’t remember the source) litvak Ashkenazim do not say tehillim on Shabbos for two reasons:
1. In general we do not make personal requests on Shabbos (as evidenced by changes in the shemoneh esrei – the gemara in brachos says it’s a burder on the tzibur)
2. Depending who you’re saying tehillim for, it could bring a person to tears and we do not cry or mourn on Shabbos as it is a day of simcha.
That being said, Sefardim (certainly Persian) and Chabad Chassidim (potentially others as well but I don’t know) say tehillim on Shabbos for cholim.