Don't be staring at me. / Don't keep staring at me. We will be singing. / we will keep singing. Do 'be' and 'keep' mean the same thing in these sentences?
"Be" and "keep" don't mean the same thing.
Don't be staring at me. / Don't keep staring at me.
"Don't be staring at me" is a less common, possibly dialectal way of saying "Don't stare at me". I would probably never say "Don't be" + gerund-participle. (I would use "Don't be" followed by an adjective or noun.)
"Don't keep staring at me" means "Don't carry on staring at me", "Don't continue staring at me" or "Don't stare at me repeatedly". Don't keep staring at me: I've already told you not to.
We will be singing. / We will keep singing.
"We will be singing" is a way of talking about the future. It sometimes refers to singing on just one occasion. Tomorrow we will be singing Christmas carols.
"We will keep singing" means "we will continue singing", "we will carry on singing", or "we will sing repeatedly". We will keep singing carols for at least a week. "Be singing" would be possible here too, but "keep singing" emphasises the continuation. If we said "Tomorrow we will keep singing Christmas carols", it would emphasise either that we intend to sing them on many different occasions tomorrow, or perhaps that we intend to carry on singing them even though some people want us to stop.