I have done this with tons of monitors. It is not an issue with anything except for older monitors that do not have built in degauss capabilities. The only thing you might ever run into is that some large monitors that already have a TON of time on them will never quite degauss perfectly in their new orientation, leaving you with a slight discoloration in one corner.
A few examples from my own current and past collection.
NEC Multisync 90 (4 or 5 years old now). No problems at any angle or orientation.
13" Mitsubishi SVGA monitor, circa 1991. Has no onboard degauss, thus it has color problems if turned.
21" Pixeltouch Kiosk monitor. I am currently running one vertical (also laid back about 45 degrees), and another horizontal (laid back at 45 degrees also). Both show very mild discoloration in one corner, but only of one color. You can't even see it unless the entire screen happens to be one color at once, AND it is THAT color (like the Neo Geo boot screen). I have another one of these that seems to degauss properly at any angle. I should also note that the discoloration has been gradually going away over time on both of them.
No name 14" monitor, circa 1995. Running vertical, laid back at 45 degrees, and decased. No issues at all. Previously ran facing straight up in a cocktail for a year.