If there are problems with updating or with the permissions then best is to download the full version and trash the currently installed version to do a clean install of the new version.

Download a new copy of the Firefox application and save the disk image file to the desktop

• Firefox 36.0: https://www.mozilla.org/en-US/firefox/all/

• Trash the current Firefox application (open the Applications folder in the Finder and drag the Firefox application to the Trash) to do a clean (re)install
• Install the new version that you have downloaded
• https://support.mozilla.org/kb/Installing+Firefox+on+Mac

Your personal data is stored elsewhere in the Firefox profile folder, so you won't lose your bookmarks and other personal data when you uninstall and (re)install or update Firefox.

• http://kb.mozillazine.org/Profile_folder_-_Firefox

And in case you did install Firefox 36.0.1 your useragent may be set at Firefox 16.0

https://support.mozilla.org/en-US/kb/websites-say-firefox-outdated-or-incompatible

marleyk76 said

I see other people have this problem and hope there is a solution. My Firefox browser is stuck on 16.0 and nothing happens when I try to update. It seems to download, but I'm still on 16.0. I have tried every which way to update my browser. Now my bank (online banking) is saying I have to update to continue with them. Please help. (computer: MAC 10.6.8)

PROBLEM SOLVED!!! and I feel so stupid, but maybe this will help others. I had downloaded Firefox 36.0 and it was in my applications folder, BUT I kept clicking on the old Firefox icon so still brought up 16.0. All this time and I just needed to bring the new icon out onto the dock and delete the old one. Thank You for those who tried to help me.

Your troubleshooting information list seems to confirm that you have Firefox 16 ("browser.startup.homepage_override.mstone": "16.0").

{
  "libraryVersions": {
    "NSSSMIME": {
      "minVersion": "3.13.6.0 Basic ECC",
      "version": "3.13.6.0 Basic ECC"
    },
    "NSSUTIL": {
      "minVersion": "3.13.6.0",
      "version": "3.13.6.0"
    },
    "NSS": {
      "minVersion": "3.13.6.0 Basic ECC",
      "version": "3.13.6.0 Basic ECC"
    },
    "NSSSSL": {
      "minVersion": "3.13.6.0 Basic ECC",
      "version": "3.13.6.0 Basic ECC"
    },
    "NSPR": {
      "minVersion": "4.9.2",
      "version": "4.9.2"
    }
  },
  "javaScript": {
    "incrementalGCEnabled": true
  },
  "accessibility": {
    "forceDisabled": 0,
    "isActive": false
  },
  "application": {
    "userAgent": "Mozilla/5.0 (Macintosh; Intel Mac OS X 10.6; rv:16.0) Gecko/20100101 Firefox/16.0",
    "supportURL": "http://support.mozilla.org/1/firefox/16.0/Darwin/en-US/",
    "version": "16.0",
    "name": "Firefox"
  },
  "modifiedPreferences": {
    "security.warn_viewing_mixed": false,
    "gfx.blacklist.webgl.msaa": 4,
    "browser.startup.homepage_override.mstone": "16.0",
    "network.cookie.prefsMigrated": true,
    "browser.cache.disk.capacity": 1048576,
    "privacy.sanitize.timeSpan": 0,
    "browser.startup.homepage_override.buildID": "20121005155445",
    "browser.cache.disk.smart_size.first_run": false,
    "privacy.cpd.sessions": false,
    "privacy.cpd.history": false,
    "browser.places.smartBookmarksVersion": 4,
    "places.history.expiration.transient_current_max_pages": 100664,
    "privacy.cpd.cookies": false,
    "privacy.cpd.downloads": false,
    "extensions.lastAppVersion": "16.0",
    "browser.cache.disk.smart_size_cached_value": 1048576,
    "accessibility.typeaheadfind.flashBar": 0,
    "privacy.sanitize.migrateFx3Prefs": true,
    "browser.startup.homepage": "http://us.mg4.mail.yahoo.com/neo/launch",
    "privacy.cpd.formdata": false,
    "places.database.lastMaintenance": 1424619568
  },
  "extensions": [
    {
      "version": "1.1a",
      "name": "Troubleshooter",
      "isActive": true,
      "id": "troubleshooter@mozilla.org"
    },
    {
      "version": "20150225.01",
      "name": "Firefox Certificate Store Hotfix",
      "isActive": false,
      "id": "firefox-hotfix@mozilla.org"
    }
  ],
  "graphics": "Troubleshoot data provider failed: graphics\nTypeError: ext is null",
  "userJS": {
    "exists": false
  }
}