IMHO opinion you can use either "know" or "learn" with "off by heart" in BE (or you can drop the "off" and just say "by heart"). Depends on the occasion.
e.g.
"I still know 'George, Who played with a Dangerous Toy and suffered a Catastophe of considerable Dimensions' off by heart ... I had to learn it off by heart in primary school."
But then I’m only a simple northern lass who has spoken the language for more than 40 years ... (I also read package leaflets very, very, carefully. ;-)
No useful dictionary entries to offer, but I’ve found a few googlies:
“Nobody learns Monk from the score - you have to know it off by heart.”
http://www.nday.co.uk/IndependentInterview.html"If you have ‘picked up’ my enjoyment of it, it must be something about the way I have written because it is the poem we have been concentrating on all the time. In fact I know it off by heart (which is perhaps not too surprising by now)."
http://openlearn.open.ac.uk/mod/resource/view..."This is your reference list, and you need to know it off by heart."
http://www.cs.bham.ac.uk/~rxb/HTML_text/hci_1..."Rude and funny. My class had learnt it off by heart by the third read through!"
http://www.amazon.co.uk/Best-Books-Primary-Sc...“Mr Straw's self-regarding sense of fun impelled him to cap that with: "My right honourable friend does not need to go to a bookshop for it, because she has learnt it off by heart as I learn her speeches by heart.”
http://www.telegraph.co.uk/news/main.jhtml?xm..."There is no real way of knowing what gender a French noun is other than to learn it off by heart."
http://www.languagetutoring.co.uk/Gender.html