Our forge cleric wanted to manufacture diamond rings and use the diamonds later for raise dead and similar spells.

Artisan's Blessing states that the end product must include some kind of metal. The metal used as material for the ritual then magically forms even the non-metal parts of the product.

Can the cleric keep using this feature to basically convert coins to diamonds?

This is not how the feature is intended

While David offers a great answer to making it work, I'm going to offer the other side of the Artisan's Blessed coin; that this clearly isn't the intended use. Magic has a component cost for a reason, and you're trying to abuse a class feature as a loophole to get around this.

If you are making a ring purely for the diamond contained within, you're not really making a metal object, are you? Once your DM allows you to create a cheap brass ring with an expensive 99.9 GP diamond in it, this feature essentially reads "you can create any mundane object costing less than 100 gp".

Need rope? "Yeah I'm making a grappling hook with 900 feet of rope, and then I'm cutting the hook off."

Need perfume? "Yeah I'm making a perfume bottle with perfume in it, with a metal stopper."

Need a chest? "Yeah I'm making a wooden chest with a metal lock on it."

While your DM obviously has the final say in all matters, and some DMs will allow this kind of thing, it is clearly breaking the intend of the feature, which is to make a metal object, not "anything you want, with some cheap metal attached."

Want to store your diamonds? You can use your artisan's blessing feature to make rings worth no more than 100 GP out of metal with an opening, and then afterwards manually add an existing diamond to it.

Yes, but it can't exceed a value of 100 gp

The Channel Divinity is fairly clear (emphasis mine):

You conduct an hour-long ritual that crafts a nonmagical item that must include some metal: a simple or martial weapon, a suit of armor, ten pieces of ammunition, a set of tools, or another metal object...

The thing you create can be something that is worth no more than 100 gp. As part of this ritual, you must lay out metal, which can include coins, with a value equal to the creation.

So, a Cleric of the Forge Domain can indeed quite literally convert metal coins into diamond rings, as long as the value of the ring is lower than 100 gp. There are no official diamond jewelry of less than 100 gp of value (the Dungeon Master's Guide lists
some types of jewelry as Art Objects in Chapter 7), so whether such a low value diamond piece can be created will be up to the GM. There is no direct precedent, but a gold ring set with bloodstones is valued at 250 gp. Bloodstones are valued previously in the chapter at 50 gp which is the lowest value diamond implicitly mentioned in any spell component list. Chromatic orb:

Components: V, S, M (a diamond worth at least 50 gp)

Presumably, a 50 gp diamond set in a material less valuable than gold (for example, silver) would satisfy the 100 gp maximum. However, whether such a ring exists is entirely speculation as there is no reference for a less than 100 gp diamond ring. So a GM is perfectly within his right to deny such a creation (not that he wouldn't be if there were some obscure precedent).

How strong is it?

Even if you could create a cheap diamond, there are only so many spells it would work with. Presumably, the diamond would be something of lower value than 100 gp (as the ring itself is 100 gp), so we're stuck with 8 possible spells:

1. 
   Chromatic orb: as mentioned, only requires a 50 gp diamond (doesn't consume the diamond, so not really relevant for the "many diamonds" abuse)


2. 
   Glyph of warding (partly): requires 200 gp of incense and powdered diamond, the latter of which could probably be made up of a lot of cheap diamonds. You still need to get the pricey incense though.


3. 
   Greater restoration: requires 100 gp of powdered diamond (like part of glyph of warding)


4. 
   Nondetection: requires 25 gp of powdered diamond (like greater restoration)


5. 
   Stoneskin: requires 100 gp of powdered diamond (like greater restoration)


6. 
   Revivify: requires diamonds worth 300 gp total, so a number of cheaper ones would qualify.


7. 
   True Resurrection: allows multiple diamonds like revivify, but the exorbitant 25,000 gp requirement, makes many cheap diamonds seem unfeasible (besides by the time you are level 17, finding resources shouldn't be an issue).


8. 
   Symbol (partly): requires an opal/diamond powder mix of 1000 gp, so the opal would have to be acquired as well.

So, for the most part, revivify is the only spell they are casting that will affect the game substantively, from my point of view (since they can make the cost up of smaller diamonds), but the rest of the time they are still paying the gold cost so abuse isn't particularly harmful to the game.

You specifically mentioned raise dead, which would not be eligible as raise dead requires a diamond of higher value (500 gp).

In conclusion: There is really only cause for concern if you expect revivify to be abused during the game to where death becomes less of a threat than you desire in the campaign. Keep in mind that revivify has a 1 minute timer, so it isn't the most effective resurrection spell especially if conflicts are designed to last longer than 1 minute.

No, the rules list what you can make, and a diamond ring is not among them

You conduct an hour-long ritual that crafts a nonmagical item that must include some metal: a simple or martial weapon, a suit of armor, ten pieces of ammunition, a set of tools, or another metal object.

The part after the ‘:’ is not "for example", but a list of things you can do.

A brass ring with a diamond is not "another metal object", nor is it a simple or martial weapon, a suit of armor, 10 pieces of ammunition or a set of tools. So you cannot create a brass ring with a diamond in it.

Because a brass ring with a diamond is not a metal object. It is an object that contains metal.

On the other hand, a brass ring is "another metal object", so it can be created.

The requirements in this sentence combine. The object must be nonmagical. The object must include some metal. And the object must be from that list.

Ammunition that contains no metal? Banned.

Now, if you can convince your DM that small diamonds with lead weights are piece of ammunition, and that they are worth less than 100 gp, then go for it.