The .358 Winchester, if I recall correctly, is based on the .308 cartridge, necked up to hold a .35 caliber bullet, much like the .35 Whelen is a necked-up .30-06.
As long as the extractor holds the cartridge to the bolt face it would ignite, but the .308 bullet wouldn't find much purchase in that .357 bore. The brass would come out formed for the .358 and accuracy would be horrible. I don't think that the rifle would be damaged, but I don't believe that it's a good idea. I can't see the utility.