The Firefox extension does not work if it is installed in Firefox's extensions directory (the so-called "global installation", scenario two in this document). I have verified this bug with HTTPS Everywhere 3.5 installed on Mozilla Firefox 24.4 ESR running on Windows 7 Enterprise 64-bit SP1.

After some research, I have determined the cause to be hardcoding of rulesets.sqlite's location. Specifically, src/chrome/content/code/HTTPSRules.js (line 372 at the moment) hardcodes its location as paths ["extensions", "https-everywhere@eff.org", "defaults", "rulesets.sqlite"] from the profile directory ("ProfD") which is not always the case.

P.S. In case you are wondering about the exact way I used to install the extension, I unzipped https-everywhere-3.5.xpi to the C:\Program Files (x86)\Mozilla Firefox\browser\extensions\https-everywhere@eff.org folder.