"Take a bite off the loaf" is perfectly fine.
It's a perfectly grammatical sentence, and it would mean that you are taking a loaf of bread and biting right into it, without cutting or tearing it up into smaller pieces first. The reason why it's not normally said is because English speakers don't usually eat bread that way - but if you bought a loaf of French bread that was intended to be eaten straight away, you might.
Conversely, if you say "tore a piece off the loaf", you're usually implied to have done so using your hands unless the context says otherwise.
As for the difference between "take a bite off of", "take a bite out of", and "take a bite off of", I would say that the differences are fairly subtle (and judging by the answers to another question on the English Language SE, possibly dependent on national dialect). "Take a bite of" can be used in contexts where you're not biting something off or out of something, like a bowl of porridge; it often refers to the consumption of a single bite of food rather than the act of biting into something. For "bite out of" vs "bite off of", they're fairly close in meaning, and do refer to the act of biting into something to eat it, though you'd take a bite out of the side of something, while you'd take a bite off of its top.