I haven’t burned any items yet, but I do know that the Self Study Quiz script makes it possible to review items that you’ve already burned (only within the Self Study Quiz script, though). You can choose ‘burned’ items and the amount you want to review in settings > items.
Other than that, it is apparently possible to ‘unburn’ burned items, so that they get added back to the Apprentice batch, meaning that they show up during your Wanikani reviews again.
On that note, I’d like to ask the advanced Wanikani users to correct me if I did get it wrong.