Start Firefox in Safe Mode to check if one of the extensions (Firefox/Tools > Add-ons > Extensions) or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox/Tools > Add-ons > Appearance).

• Do NOT click the Reset button on the Safe mode start window or otherwise make changes.

• https://support.mozilla.org/kb/Safe+Mode
• https://support.mozilla.org/kb/Troubleshooting+extensions+and+themes

• http://kb.mozillazine.org/Uninstalling_add-ons
• http://kb.mozillazine.org/Uninstalling_toolbars

Thanks for replay, but I don' t want to uninstall or disable this plugin, I just want to get rid of its toolbar (hide it) so I need an advice from some extension developer how to modify the source of this extension to achieve that.

Have you tried contacting the developer of that extension?

Beyond that, it would be nice if we knew the name of that extension.

Yes. Developer of this extension is company where I work, I have talked to them but they don't want to enable possibility to hide the toolbar.

This is a restartless extension, so you can easily disable and enable in the Add-on manager (possibly via a pinned tab) to hide that toolbar when you do not need it.
It also looks that the extension is signed, so modifying it would invalidate the extension.

I already modified some part of it and it still works. The toolbar of this extension contains just a useless one button and company logo. There are other functions which I still need to use (for example data sending to server, signing in/out of company website, etc.).

I can't modify toolbar in Customization mode.