I agree with Soundshift's post #2. Perhaps this is just a BE usage?
They sound unidiomatic because the expression is "to learn something" or "to learn something (off) by heart". If you choose to include the "off", you need to follow it with "by heart". "To learn something off" is not a possibility. So:-How do these sentences sound to you?
"I've got to learn these words off in a very short time."
"You are to learn these two pages off by tomorrow morning."