Can I run a 4400 without a fan ?

Only for a few minutes - booting to windows - no 3d stuff. I am trying to
trace where a high pitched whine is coming from. I know it isn't power
supply as the noise disappears in standby while the fan keeps going.

I can safely unplug fan for intel cpu for a little while as it has big
heatsink.

So its the video card or maybe hard drives. I need enough time to put the
case lid back on as the noise is quite different.

It would a lot simpler to just stick your finger in the fan while its
running and see if the whine goes away.

I would go with the finger stopping the fan trick, but if not then the answer
to your question I would have to say is YES. What would happen if someone were
using their video card and the fan died and they didn't know it? How could they
know the fan died with the case closed? When you boot into windows the gpu gets
hot, but not like it does from doing 3D. That doesn't mean I would run the
system for long without the fan, but I don't think few minutes will hurt it. If
anything, if it gets too hot your system will probably lock up, then the gpu
isn't doing anything so its not radiating heat.