I think this could get mighty complicated:
What kind of long-term damage are we trying to avoid? If used and stored indoors in a climate controlled environment, temperature (and rapid changes) and humidity (and rapid changes?) aren't an issue. The other culprits are UV light(sunlight or manmade), oxygen(especially ozone!), and migration of plasticizers.
So we want the substance we apply to form an oxygen barrier, prevent plasticizers from leaving(and/or replace them), and block UV light if necessary. Perhaps suntan lotion would work? I've used it on automotive vinyl for these reasons, FWIW. Even then, dozens of formulas to choose from. Different mink oil formulations as well, as mentioned above.
What might be the negative effects? If the additive softens the surround, it might change the compliance(Vas). The additive could harm the adhesive used to glue the surround to the basket or diaphragm. Or it may cause the surround to swell, dissolve, or weaken to where it's easily damaged. One substance that seems universally despised for all plastics and elastomers is Armorall!
The Sunfire subs are around 15 years old now. Has anybody heard of a cracked, dry rotted, or otherwise failed surround? I'd guess hardly anybody applied mink oil to their surrounds as instructed. Maybe Sunfire or Bill Flannery have some data on failures and if mink oil helped?
If we knew for sure *what* the surround(and adhesives) and particular "mink oil" consisted of, we could make some better guesses or even experiment.