Alongside billiards, leathercraft is another primary hobby of mine, so I'll leave a few thoughts here. You can find some of what I work on
here if my credentials/knowledgeability are suspect.
Even in the leathercraft world, the realm of oils/conditioners is incredibly hazy; there are few facts (largely due to lack of repeatability/generalizability and cellulal-level analysis), and a lot of myth/folk remedies floating around. Mink oil and cooking oils are well-known one to avoid from the perspective of it seemingly achieves the goal of making leather supple/restored from a tactile perspective because it softens it. This extends to plenty of other oils. If it satisfies and doesn't go rancid, great. Is it absolutely the best choice? The answer ranges from no to not really, but there's insufficient evidence from an organic chemistry standpoint. Happy with the results is different from beneficial material interaction.
Given that every tannage of leather is its own chemical concoction, it'd be impossible to generalize a panacea for restoration. Neatsfoot oil and extra virgin olive oil are more accepted oils, but I don't want to indicate that these are the ones to default to; neatsfoot really only belongs in the world of saddlery and tack where it's hard-worn and spends significant time in the sun, less so your wallet made from a thinner version of the same stuff. EVOO is thought to be a decent way to go for processes that dry out the oil, such as wet molding (e.g., old-world/period reenactment armor, gun holsters).
Oils and conditioners are typically concerned with keeping the top grain intact and from cracking, which is not what we deal with in cue tips. We're using the fibrous grain structure, so anything overheard about an application to leather cannot be carried over here. Structurally, a suede restorative product is more applicable, but that is almost always a chrome tan process rather than the vegetable tanning we see in tips.
That said, oils and conditioners are overprescribed. They immediately feel like they're doing a lot because they soften and darken the leather, giving the sensation of a positive change. In general, cue tips or not, leather goods are safer erred on the side of not treating it. Specifically in the case of tips, I don't see a benefit to the upsides of conditioning, as the resulting softness could "dent" the tip if gets too soft, somewhat like an Elkmaster installed on a snooker cue with immediate mushrooming to one side because the test hit was off-center. As echoed elsewhere in this thread, you certainly wouldn't want to do anything to a layered tip since the adhesive would be degraded.
The aforementioned way of impregnating a tip with oils through continued contact also doesn't hold in practice, as we only ever get "patina" at the surface level. Once you trim excess off of a tip, you've gone past the penetration depth of any oils transferred through casual exposure. I do experiments with my leather scraps for patina such as leaving it out in the sun or exposing leather that shouldn't get wet to water drops, and I've never found anything besides slathering a liquid on to penetrate any deeper than the absolute 1oz (0.25mm thickness) top of the leather. On the other far end, soaking it in oils would result in full penetration, but that risks the whole integrity of the fibrous grain that isn't being held by a binding top grain, so not recommended.
Similarly, milk duds are immersed in a fluid, but the end result is a drying out effect closer to soaking it in water than it is oil; the end result is a harder tip that doesn't mushroom as much. I can't confidently speculate too much on the milk casein - leather interaction, but I'm suspicious about its benefits at best since it's a stabilized product through the tanning process and rancidness would be a known, larger risk. All that said, I've enjoyed 8/9 of the ones I've used from pooldawg8 (free plug!).
