Do entrapped nerves in the face, head, or upper neck area cause humming sound like the bee buzz?

No: If you have tinnitus, you need an audiogram and an ENT examination.

Tinnitus: Normally a humming sound or the high pitch machine noise or gear noise is do to a problem in the ear or eustachian tubes. It is called tinnitus (tin a tuss) and is sometimes attributable to medication such as aspirin. You should see an ENT doctor for evaluation and possible treatment.