No, usually a tethered jb means you won't be able to use anything that uses mobile substrate (like Bite and many other jb apps). On top of that, just having those installed means you might not be able to boot without connecting. If you DON'T have ANY mobile substrate apps you might be able into a state that lets you use your phone, just without any jb apps/capabilities.
See this, from the Dev Team:
“Tethered” does not*mean you cannot boot at all without PC/Mac assistance. *If you have not installed any tweaks that hook into important programs like SpringBoard or CommCenter, your device will actually boot. *However, jailbreak programs like Cydia won’t work (and Cydia may still have a white icon). *Also, certain built-in apps that had to be moved by Cydia will fail (Safari being the most noticeable example).
If you’ve installed MobileSubstrate tweaks that hook into SpringBoard or other important programs, your boot will actually fail (you’ll get stuck at the Apple logo). *You need to use redsn0w to “Just boot tethered right now”.