From my reading of the code, this whole module doesn't work if there is no writable directory. The so-called "texcache" is not really a cache at all, it's also a working directory, where intermediate files are generated and processed with dvipng. So even if we assume a system that does have a TeX installation and the ability to launch external commands, but does not have any writable file system, we won't be able to run these commands. (This isn't a simple matter of disabling caching.)

Correct me if I'm wrong about that.