I don't *think* I suggested oiling canning jar rings... might have suggested it for canned goods (in cans) or possibly even using vaseline on the jar lids if they must be stored in a very damp area.
Anyway, you can remove them any time after the jar is cool. In practical terms, I usually take them off after letting the jars sit overnight.. I'll also clean the jars at that time if necessary.
What I do with jar rings (and dear Lord, I have a TON of them. I think I'm on my 6th Sterlite container of them!) is clean them in warm, soapy water. I set them out on a towel after rinsing, and then stack them on a broiler pan or other pan with at least a shallow rim. I heat the oven to 200°, and put the pan of rings in for 30 minutes. This dries them *thoroughly*, and allows them to store without rusting.
I then put them in Rubbermaid type containers (I'm using some which are maybe 25" long, 16" wide and 6" deep? Very approximate, as I've run up and down the stairs about three hundred times in the past 2 days, and I'm not doing it again to measure them tonight! LOL) which seal well enough to exclude moisture. If you want, you can put them into heavyweight ziploc bags (although those are SUCH awful quality these days, I hesitate to recommend them for anything important).
If you have a nice, dry area for storage, you could string them on lengths of string and hang them along a wall or from the rafters. My cellar is a bit too damp for that.
Toss out any which have gotten bent at all... it's vital for a good jar seal that the rings press evenly and firmly on the lids during the canning process. Also toss any which have gotten rusty enough that they are difficult to screw onto the jar, for the same reason. Every once in awhile, it seems I'll get a bad batch in a case of new canning jars. That's not a real problem, given my crazy inventory of the things (and boy, if you have some like I do from 40 years ago- or longer- you can REALLY see how cheap they've gone on the newer ones!), but it's frustrating anyway. Quality control just doesn't exist these days.
Summerthyme