Why does a relay have a minimum applicable load?

The primary reason that almost all relays have a minimum load requirement is that the mechanical action of closing coupled with an actual current flow are required to 'whet' the contact and break through a layer of oxidation that invariably builds up.

That is one reason that small signal relays generally use expensive contact alloys which resist oxidation, but as the phone company found out decades ago, even pure gold contacts can have issues in a high humidity environment. While oxidation doesn't affect the gold contacts, repeated cycles of moist/dry air would deposit an insulating layer.

Mercury wetted (reed) relays are used where you cannot guarantee minimum load currents. These have a film of mercury on the contacts that will make contact even with no current as it forms a liquid junction. Generally the contact resistance will be slightly higher than the best gold contact resistance. The switching lifetimes may be much higher that other types. The load currents are generally lower as they are only usually needed for small signals.

In your case where the live loads are of a respectable value almost any dry relay type will work reliably.

The longevity of a dry contact will not appreciably change even if no current flows but age and lack of load current will slowly increase the low current contact resistance as mentioned by others. This means that marginal low current signals will suffer higher than specified contact resistance until such time that an adequate current has been switched.

EDIT:
It is also possible for the thin oxide layer that forms on the relay contacts to cause the low current resistance to increase and remain high until reasonable load currents have been switched that will 'burn' it away.

Relay will work at any load (below maximum of course). However if you use less than minimum permissible load, it may fail before it should (less switching operations). The reason for this being after time some corrosion can get on the contact. Minimal current is measured/calculated to be able to break through contact corrosion and it also slows down the corrosion process itself.

You can use the relay with less current, but if you are using it in a device that needs high reliability or you can't change the relay if it fails, it's not a good idea.