Do Coin Albums Eventually "Ruin" Copper Coins?

I have never considered using press - in cardboard albums. I think they are a no-no, especially in the way I think about them.
Why?
If the coins have often been pressed into the holes with ungloved fingers, there is a big risk of fingerprint stains on the coins, and especially with untoned bronze coins.
Some albums have residual chlorine in the cardboard from the fact that Cardboard is often made from recycled chlorinated paper, as one of the ingredients.
Some of the glues used in the manufacture of cardboard albums, have a high sulfur content.
The gassing out of oxidizing agents (principally chlorine and sulfur based compounds), present a risk to the chemical attack (albeit slow, BUT unnoticed, due to long term storage), of the coins.

Anther problem with cardboard albums, is that the design of some of them allow the collector to see one side of the coin only.

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Coin album pages made from PVC free plastic offer virtually NO risk to bronze and copper coins in long term storage.

Disagree, sorry. I would never recommend that.

Clear acrylic laquer protectives (a typical example being nail polish), should have little or no injurious effect on bronze coins.
However, two problems arise:
1. The chemistry of the nail polish may vary from product to product, and probably contains other ingredients, of which the chemistry is unknown.
2. Nail polishes stick very well to nails, but not necessarily to the metal surface of a coin. The polish is inclined to chip off in tiny areas, leaving those areas differentially unprotected. This may, over time, result in blotchy areas, some toned, and some untoned.

A non nail polish clear spray acrylic lacquer may be OK, but the hardened lacquer still has to be protected, perhaps in a hard clear screw sealed acrylic capsule.
The upshot of all of this is that you are probably better off using a hard plastic acrylic capsule only.
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Hardened acrylic lacquer is very easily removed with acetone.

Where you live will play a role as well.

Climate-controlled safety-deposit boxes are expensive. Most MS coins, especially in slabs, will not react to normal climate variations. Even fewer AU coins, if any. Think it through.

Original post says both Albums and Folders. Please note the differences between the two. Folders are basically open to the air. Albums have at least a plastic slide covering the coins. I have been using Whitman Albums for a long, long time. I keep them in a Zip Lock bag and no problems of any kind yet.

I see what you mean.
I was wondering about this.
These coins have been in this folder since the early 1960's.

The ultimate answer is "yes", some coin albums can and do ruin the coins they were theoretically designed to help protect.

As others have noted, the "push-in" folder type of album has two main flaws:
- They're made of cardboard, which often has a high sulfur content, so the folders themselves can emit gases as they age which react with the coins, especially silver and bright shiny copper.
- They're not airtight, so allow exposure to the atmosphere and whatever conditions the atmosphere are like. High humidity and high sulfur content can both easily access a coin in a cardboard album.