Edit 1/7: I'm still not sure of the cause, but I have been able to re-enable code analysis by using the ReSharper_EnableDaemon keyboard shortcut Ctrl+Alt+Shift+8 with non-working files open. Curiously, toggling a working file off adds it to the Code Analysis Exclude list, but there are no other files in the list, despite some clearly acting as though they have been.
Resharper is no longer doing any syntax highlighting. I have a number of style rules around variable/method naming, using explicit types instead of "var", etc. Other things like warnings around possible null exceptions are also not being shown.
Other features of Resharper seem to be working normally (build, Ctrl+T search, Alt+Enter for quick operations, etc.).
I've tried digging through the options looking for something that might be causing the issue but haven't found anything (toggled off, saved, toggled back on, saved) that has fixed my issue.
I did perform a Windows Update yesterday, the timing suggests that this is related. However, I've been unable to find any Visual Studio settings that fix my problem either.
After some more digging I've found some files are working properly, with all the code analysis and syntax highlighting and style rules I expect. The "Elements to Skip" section of "Code Inspection > Settings" only has "*.min.js" in it, but MANY other files are being ignored.
Edit: VS 2015, R# Ultimate
Edit 2: I've deleted the Resharper transient folders and uninstalled/reinstalled Resharper: no dice.