I don't think that we can tell you much without a screenshot. It could be one of any number of issues. But you could also add some clarifying detail, as I'm not really sure if you are opening old CS5 files in CS6 and experiencing character drops, or if you're placing new text into a new file, or something else. What kind of files do you have? I mean, are you talking about InDesign files, or client-supplied Word files, or something else? When you say that some of the glyphs "don't work anymore", exactly what is going on? Pink boxes? Pink highlighting with a different glyph? Incorrect glyph variants? Sudden appearance of Trad glyphs where there was simplified previously? Glyphs holding up picket signs and marching around the pasteboard?
When I open the same file in Indesign CS5 and Indesign CS6 ont the same computer with the same fonts activation, the result is not the same.
View from Indesign CS5
View from Indesign CS6
There is some glyphs replaced by Pink highlighting squares.
When you copy/paste a glyph in the "search panel", usually in Indesign CS4,CS5 the reference of the glyph appeared (ex; <631F>) but in CS6, you'll see the glyph. Perhaps something weird with the code of the font.
My recollection is that IDCS4's search box sometimes displays characters rather than bracketed codes, but I never figured out why. As for MHei not working, I notice that the Linotype website gives different character counts for different weights of MHei, which could account for some "pink boxes." If that's not the issue, then my next step would probably be to zap Adobe's font caches.
Immediately after zapping those font caches, I'd try turning off whatever font management utility you're using:
the same computer with the same fonts activation,
and just making sure that the font is installed directly into one of those font folders that InDesign can read.