If you wear armour, you need to have other things on underneath (such as a leather tunic) to prevent it chafing, or cutting into you if someone inflicts a heavy blow on you (I know this from a number of years experience of doing battle reenactments). So if Talking Beasts wear armour then they should also wear other clothes, at least for this purpose.

Would the fur guard against chafing, though, King Erlian? I suppose it would depend on how thick and dense the fur is.

I think if the animals did wear clothes of any sort, they would probably be tunics or other basic medieval wear; very simple and peasant-like. That type of clothing would also probably work well underneath armor.

The tailors and weavers of Narnia would likely have been fauns and dryads and naiads, I'm guessing, since I think it would be pretty difficult to sew and weave fabric without the use of opposable thumbs.

It seems like the animals in Narnia did not really have a sense of "nakedness" like humans do, so they don't need clothes for that reason. I can't really remember any time when animals wore clothes, except when Shift was trying to convince the Narnians he was a man.
Unless the Narnians needed clothes for a specific purpose such as armor or maybe a coat when they were freezing, I don't think they should have clothes on. Perhaps a sash or something to signify a title, like Reepicheep's gold band, but I wouldn't put them in complete outfits.

Wasn't Reepicheep wearing clothes in both the films he appeared in?
Edit. (Or maybe it was only in Prince Caspian)

In Magician's Nephew I wouldn't say so but yes for Last Battle. I would love to see most of the animals in armor if they do make a film of LB.

I prefer to see the Talking Animals wearing accessories rather than full-on clothes--it helps to visually distinguish between the Narnians and the non-talking types we also see.

The tailors and weavers of Narnia would likely have been fauns and dryads and naiads, I'm guessing, since I think it would be pretty difficult to sew and weave fabric without the use of opposable thumbs.

Doesn't Lewis make a comment to this effect while Shift is sewing in LB?

I don't think they should wear full on clothes (except in the case of armor and other practical things like that). I like Mel's suggestion of having accessories.
After a quick google search it looks like Reepicheep didn't wear clothes, except for some simple armor during battle and a brown strap of some sort. I imagine it has some practical use, but I can't seem to recall what it is at the moment.