There are a couple of potential reasons to perhaps avoid saying Mi Sheberachs for Cholim or people who get Aliyos during Leining on Shabbos (Tircha Detzibura and Pesikas Mammon). However, whether due to Minhag or otherwise, many Poskim are lenient (see above links), and allow such Mi Sheberachs to be recited.
I am looking for sources in Halacha that directly prohibit saying either one (or both) of these Misheberachs on Shabbos for one of the two reasons above.