When to replace FS bearings? I know that when a fork stops working plushly, it should be serviced in order to escape permanent damage. When a chain is noisy, it should be oiled, and when it is worn - replaced - again to prevent damage.\

But what about the bearing in a full-suspension mountain bike frame? How do I tell if mine should be replaced? What calamity could result in not replacing them?
First, not all full suspension bikes use bearing for all pivot points, and some bikes don't use them for any. They use bushings instead.

That said, assuming you have bearings at all points on your bike:

There are 2 types of bearings damage which require replacement in a suspension system. 

The first, bearing play, means lateral movement inside the bearing cartridge itself. This occurs when the ball bearing wears deeply enough into the bearing race that there is space between the 2 races which is larger than the ball itself. 

In a suspension system, this often presents as clicks or creaks in your suspension. It can be mimicked by loose pivot bolts, so the first check is to remove, grease, and re-torque all the bolts in your system. If the play persists, you may need to replace bearings.  

The second, rough bearings, is often caused by extended use, or heavy impacts in the course of the use of your bike. 

The ball bearings or races get physical chips or divots in their surface called pits, and thus no longer roll smoothly. 

The presentation is often the same as with bearing play, clicks or creaks in the system. 

The good news is that the diagnostic for both types of bearing issues can be done in one go. 

Once you have pulled, greased and re-torqued the bolts in your system, disconnect the shock from you system, remove your rear wheel, and move the swing arm through its travel by hand. You should be able to feel any roughness in the bearings. 

Repair in both cases requires replacing the bearings. This sometimes requires or at least strongly recommends tools specific to your brand or model of bike.