I've not heard this particular usage before, but I suspect it's US rural/dialectal. There are many versions of "I don't give a fig/damn/toss/monkey's/rat's arse/etc.", where the primary "meaning" is that the thing you don't "give" is something worthless/inconsequential. So here it could be OED's definition #2...
rip - A handful of the stalks of unthreshed grain or hay, esp. as a decoration or sample; (also) a single stalk of this kind
...or (given that the above is primarily Scottish/Irish), perhaps it's more likely to be definition #6...
rip - A worthless or worn-out horse; one that is of inferior quality. Also occas. applied to cattle.
Of course, any individual speaker may have no idea exactly what he's "not giving" - he may simply think of it as a meaningless euphemism standing in for one of the more common profane alternatives.
Unless there's a somewhat contrived context contrasting negated/non-negated usages ("He doesn't give a damn about me!" - "Forget him! - I give a tossdamn!") you wouldn't normally try to adaptapply this idiomatic usage to non-negated contexts.