No experience with refinishing but my wooden cutting boards and wooden knife handles are in very sad shape. For several weeks now, I have been thinking of trying to help them both for both hygienic and esthetic reasons so I have done a slight amount of research. This is one of the more reasonable sites I visited
https://www.lumberjocks.com/threads/cure-time-for-a-pure-tung-oil-touch-up.355787/
What I know from previous research on paints is that tung oil is heated to almost the fully polymerized stage so that when you apply it, the oil will react with oxygen and fully polymerize, ie, harden, cure. A finish is intended to soak into the wood grain and a paint is intended to remain on the surface. Both are expected to cure so they don't transfer to your hands when touched.
Therefore, it would seem reasonable that any applications to cutting boards and/or knife handles would try to enhance the finishing technic. Both time with oxygen and heating in the sun would be appropriate as suggested by one of the folk on the woodworking thread.
I don't know about the toxicity aspect but one of the major domestic tung oil producing sources was a small town halfway between Tallahassee and Perry FL. The trees were beside Hwy 27 and tourists would occasionally ignore the signs and try to eat the nut/fruit with uncomfortable consequences.
The thread above emphasizes that you should read the label to ensure the oil does not have any "dryers" (chemical additives that may include heavy metals) for impatient painters.
I watched a video for treating knife handles that compared mineral oil, tung oil, and a couple of "wood treatment" products. The tung oil was a clear winner. Have not seen any videos comparing cutting board treatments yet.
Might try "curing" a tung oil treated knives and a cutting board in a solar oven. I never get around to using it for anything else.