"Another program on your computer would like to modify Firefox with the following add-on" - how to disable?

Question

I am using the latest Firefox (28.0) on Windows XP in a shared computer environment. Although I do not have Administrator access, I installed Firefox myself and can change whatever Firefox config settings I want (including about:config and mucking with the files in the installation directory).
I want to disable the tab that appears on Firefox startup, "Another program on your computer would like to modify Firefox with the following add-on". This tab usually appears once per day (the first time I startup Firefox each day). I want the tab not to appear at all, from now on.
The "another program" in question is some program that is used in the facility where I'm using the computer, so I can't just uninstall that program. Although this is a shared computer environment, they don't "wipe" the machine every day, just reboot. So the Firefox I installed is still there with all my config settings intact. As far as I know, I'm the only person who uses Firefox on this computer.
Solutions that don't work:
"Just click Continue" -- but then I still see the "Another program" tab every day, and don't want to see it.
"Click Remove in the Add-ons list" -- I removed the Add-on in question but the tab still appears every day.
I would like a solution that either eliminates the Add-on request for that particular application, or eliminates all such requests ("Another program...") regardless of what application requested it. The ideal solution would be if the request itself had a checkbox for "never ask this again" but, alas, no such luck.

Answer 1

Gosh... I spent about an hour preparing a response, then accidentally did control-something or alt-something which took me back a page and I lost everything I typed (even after I clicked Forward to try to get back to where I was). If anyone knows of a config setting like "don't ever trash an entire page of user input unless the user explicitly confirms it", or, "keep form input data even if the user goes Back or Forward", I think that would be a great feature.
OK, back to the original question...
I appreciate the responses thus far, but I think there have been some misunderstandings. Maybe a multiple-choice version would help:
Question: Is there a way to suppress the appearance of the notification tab that says, "Another program on your computer would like to modify Firefox with the following add-on" (meaning, any request from another program to install an add-on is to be silently ignored and discarded)?
Answer:
(A) Yes, you can suppress that notification (please state how).
(B) No, but you can request that feature (please state where).
(C) No, you can't do that or request that feature.
That's basically my question in a nutshell. If the answer is (B), I might envision a boolean about:config setting like "extensions.allowProgrammaticAddonRequests" that defaults to "true" and can be set to "false" manually. I think that would be the most direct solution (if it were possible).
I will respond to some comments from above:
"I don't understand why you are using a mozilla.cfg file to lock those preferences when you can modify the preferences for your current Firefox profile via about:config" -- This is because the "other program" in question (iCafe Manager) was setting certain config options (browser.startup.homepage, and browser.newtab.url) on every reboot (which happens every morning at that facility). I got tired of manually changing the settings back every day. Using the above-mentioned files, I successfully defeated iCafe Manager's daily interference with those config settings. The other things (like stuff related to Socks tunnelling) I probably could have just set once in about:config but doing a lockPref seemed to do no harm. The settings for "enabledScopes" and "xpinstall" were to see if they'd help with the problem I originally asked about, but they did not.
"I would look in about:config to make sure those preference modifications are in effect" -- yes, I confirmed that the settings are in effect and locked. That much works as expected.
"A) removing the iCafe Toolbar extension from the Firefox Addons -> Extensions panel results in the "Another program" tab at startup" -- true, but I think it would be more accurate to say: while the toolbar extension is not installed (due to either having been removed, or, never having been installed at all), the "Another program" tab appears on at least the first startup of Firefox after every reboot.
"B) when you allow the installation it shows up as "(disabled)" in the Extensions list but the unwanted toolbar still shows up (correct?)" -- yes, correct.
"You could explain the problem to the system administrator of your facility" -- unfortunately, there's no administrator on site, only a person who makes sure the lights are on. They'll call a tech if there's a catastrophic problem (like the net connection goes dead). Since I installed Firefox myself, and since iCafe Manager is a program they installed, I don't think they'll go out of their way to help me with Firefox (especially if I'm trying to defeat iCafe Manager's attempts to interfere with Firefox). From their point of view, their systems are working just fine.
"allow the installation and then see if you can go to the Firefox Customize menu and HIDE the toolbar as a workaround" -- since I'm not in that facility now (and probably won't be again until this summer or so), I can't experiment with those systems now. But I do recall clicking just about everything I could, including "View - Toolbars" (which seems to be the same menu as when I right-click on a toolbar line). That menu allows hiding toolbars, but from my recollection, the offending toolbar was not displayed in the list, so I could not hide it that way. Also, the Customize menu ("View - Toolbars - Customize") seems to allow only modifications to a toolbar, and not a way to hide an entire toolbar.
"it might help if you could post a new troubleshooting information list after you allow the iCafe Toolbar installation" -- ok, that troubleshooting report is included below.
Thanks again for your help.
Application Basics
Name: Firefox
Version: 28.0
User Agent: Mozilla/5.0 (Windows NT 5.1; rv:28.0) Gecko/20100101 Firefox/28.0
Crash Reports for the Last 3 Days
All Crash Reports
Extensions
Name: iCafe Manager Toolbar
Version: 5.2.0.6
Enabled: false
ID: {C058FE28-1E07-4FD1-8092-046F8A964D12}
Important Modified Preferences
accessibility.typeaheadfind.flashBar: 0
browser.cache.disk.capacity: 358400
browser.cache.disk.smart_size.first_run: false
browser.cache.disk.smart_size.use_old_max: false
browser.cache.disk.smart_size_cached_value: 358400
browser.newtab.url: about:blank
browser.places.smartBookmarksVersion: 6
browser.sessionstore.upgradeBackup.latestBuildID: 20140314220517
browser.startup.homepage: http://www.google.com/ncr
browser.startup.homepage_override.buildID: 20140314220517
browser.startup.homepage_override.mstone: 28.0
dom.mozApps.used: true
extensions.lastAppVersion: 28.0
gfx.blacklist.direct2d: 3
gfx.blacklist.layers.direct3d10: 3
gfx.blacklist.layers.direct3d10-1: 3
gfx.blacklist.layers.direct3d9: 3
gfx.blacklist.layers.opengl: 3
gfx.blacklist.stagefright: 3
gfx.blacklist.suggested-driver-version: 6.1400.1000.5218
gfx.blacklist.webgl.angle: 3
gfx.blacklist.webgl.msaa: 3
gfx.blacklist.webgl.opengl: 3
network.cookie.prefsMigrated: true
network.websocket.enabled: false
places.database.lastMaintenance: 1398414928
places.history.enabled: false
places.history.expiration.transient_current_max_pages: 53428
plugin.disable_full_page_plugin_for_types: application/pdf
plugin.importedState: true
plugin.state.npgoogleupdate: 1
privacy.sanitize.migrateFx3Prefs: true
privacy.sanitize.sanitizeOnShutdown: true
storage.vacuum.last.index: 1
storage.vacuum.last.places.sqlite: 1396778909
Graphics
Adapter Description: Intel(R) G33/G31 Express Chipset Family
Adapter Drivers: igxprd32
Adapter RAM: Unknown
Device ID: 0x29c2
Direct2D Enabled: Blocked for your graphics driver version. Try updating your graphics driver to version 6.1400.1000.5218 or newer.
DirectWrite Enabled: false (0.0.0.0)
Driver Date: 11-3-2008
Driver Version: 6.14.10.5009
GPU #2 Active: false
GPU Accelerated Windows: 0/1 Basic Blocked for your graphics driver version. Try updating your graphics driver to version 6.1400.1000.5218 or newer.
WebGL Renderer: Blocked for your graphics driver version. Try updating your graphics driver to version 6.1400.1000.5218 or newer.
Vendor ID: 0x8086
windowLayerManagerRemote: false
AzureCanvasBackend: skia
AzureContentBackend: cairo
AzureFallbackCanvasBackend: cairo
AzureSkiaAccelerated: 0
JavaScript
Incremental GC: true
Accessibility
Activated: false
Prevent Accessibility: 0
Library Versions
NSPR
Expected minimum version: 4.10.3
Version in use: 4.10.3
NSS
Expected minimum version: 3.15.5 Basic ECC
Version in use: 3.15.5 Basic ECC
NSSSMIME
Expected minimum version: 3.15.5 Basic ECC
Version in use: 3.15.5 Basic ECC
NSSSSL
Expected minimum version: 3.15.5 Basic ECC
Version in use: 3.15.5 Basic ECC
NSSUTIL
Expected minimum version: 3.15.5
Version in use: 3.15.5