There aren't really lists of models at risk, but since you said "collection," there might be some easily identified TYPES of firearms which might not be safe to fire with modern proofed ammunition.

Best bet is to take it to a professional gunsmith, or contact a local NRA firearms instructor, to have them take a look at what you have and give advice on which might fall into that category.

Spitballing a bit - Damascus steel barrels used in very old shotguns are much weaker than modern barrels (modern in the relative sense), so that's usually the most common "don't shoot this 'cuz it might blow up" type of firearm most people will handle in their life. These barrels were made by wrapping straps of steel around a mandrel and hammer forging - hence Damascus - which leaves them much weaker than modern bored barrels. Another common mode would be Springfield Trapdoor rifles in 45-70. Given a good inspection, they are typically safe to fire, but only with standard low pressure rounds, and not with higher pressure rounds intended for more modern rifle designs.

If you'd post up a list of the makes and models, some of us might be able to help sort this out with you. The challenge there being some of the older models which would be most at risk may not have brand and model sufficiently marked.

However, if there was some injurious event which happened to any of them, even a like new, modern model may not be safe for use. So if you don't figure out any model which might be specifically not up to snuff, you may want to have your son pick out which he really wants to shoot, and have it checked out at a smith.