This article incorporates analysis from mischiefmaker and DrFlux.
Dominion: Dark Ages
By itself, Ironmonger is surprisingly strong. Solo play suggests that straight Ironmonger-Money is about equivalent to Smithy-Money: 14-17 turns to get to 4 Provinces and some Duchies.
In a more diverse deck, Ironmonger is sort of like three different cards, depending on what it hits:
Treasure: +1 Card, +1 Action, +$1, could discard Copper. This card is like a Lab in a way: you are drawing the Copper and discarding it, but you still get the +$1. If you aren’t drawing Copper, then it’s about equivalent to Peddler, but with some info about your next card. On its own it’d probably cost about $4-$5.
Victory: +2 Cards, +1 Action, can discard a Victory card. This is strictly better than a Laboratory, since it’s sort of like +3 Cards / +1 Action with the discard. So on its own it’d cost at least $5.
Action: +1 Card, +2 Actions, could but probably won’t discard an Action. This is a Village, and probably not enough bonus on top of the Village to differentiate it from a typical vanilla $3 Village.
In a vacuum, Ironmonger is at its best when it’s discarding Victory cards, and at its least helpful when it is hitting Action cards. But of course, a Village is a Village, and if you need a Village you’ll still be happy to pay $4 for it.
The key, though, is that although Ironmonger never hurts your deck, you shouldn’t try to set expectations for it. If you read “Ironmonger is at its best when it’s discarding Victory cards” and conclude that you should do nothing but buy Victory cards and Ironmonger, well, you run into the same problem you do with Crossroads: you are constantly drawing a lot of cards, but they are just all green cards anyway! And if you try to use Ironmonger as a Village, it’s not really as reliable as just a simple $3 vanilla Village.
What Ironmonger really is is a kind of weakened-but-more-reliable Tribute. You can’t really count on what it’s going to do for you, but you can count on the fact that it won’t screw you over. No matter what you’ll get something decent.
Ironmonger and dual type cards
Of course, like Tribute, Ironmonger gets way better with dual-type cards.
Victory/Action: +2 Cards, +2 Actions
Victory/Treasure: +2 Cards, +1 Action, +$1
Both of these are strictly better than Lab.
Ironmonger and opportunity cost
So Ironmonger is always good for your deck. That means you should always be buying it, right?
Not really. Consider Spy — it’s a decent card, can’t really hurt your deck, but you won’t win the game buying nothing but Spies. Like the Looking Glass, you can’t just stand still in Dominion: if you aren’t moving towards the finish, you may as well be moving backwards. The opportunity cost of Ironmonger is passing up a chance at Bridge, or Throne Room, or some other powerful card that is pushing you towards endgame.
That being said, Ironmonger tends to be a bit better than most other spammable cantrips, because early in your engine build, Ironmonger is likely to hit Copper and Estates, cycling your deck a bit and giving you the more powerful effects; later, as you trim your deck, set up filtering, and/or increase action density, Ironmonger gives you the +Actions you need.
The basic takeaway, though, is that Ironmonger in an engine works much better with a gainer, so that you’re not giving up the opportunity cost of buying a different $4.
Works with/Conflicts with
In addition to the above, Ironmonger works well with cards that care about the top card of your deck. Mystic is the obvious choice but there may be others (Lookout, perhaps).
Like Tribute, Ironmonger does not work well with Cursers, since hitting a curse reduces Ironmonger to a straight cantrip (which is worse than Vagrant, a $2). It’s a mixed bag with Looters; hitting Ruins is a Village plus Ruins filter, which isn’t bad for a $4 but it isn’t great. More important is that in most cases you’re likely better off just going for the Looter (Marauder/Cultist) or ignoring it (Death Cart, unless of course there’s a Death Cart enabler on board) than comboing it with Ironmonger. That being said, Ironmonger is a strong choice if you are building an engine that ignores the available Looter attacks.
Dominion Strategy 
I don’t really understand the interaction with Lookout. If you play Lookout first, then any cantrip can replace Ironmonger. If you play Ironmonger first, in the early-mid game you are probably going to play Lookout anyway even if you turn up a good card.
Cartographer before Ironmonger seems much stronger, especially if you draw a Victory in the cartographed cards (leave it in 2nd position and Iromonger becomes a Lab).
That critique sounds correct to me. It’s the Wishing Well Problem on a different card.
Would it make sense to add Workshop/Ironworks/Armory to the “works with” list? They reduce the opportunity cost of acquiring lots of Ironmongers. (Ironworks also works great with multitype cards — like Island, although those probably won’t be sticking around long enough for you to repeatedly Ironmonger them.)
Ditto Haggler, Border Village, and University, although since those aren’t capped at $4, you may have better options for the gain than Ironmonger.
Also, I third the criticism of Ironmonger/Lookout. Lookout can set up your top card, but Ironmonger actually cares about your *second* card, like Wishing Well.
A few more speculations:
Is it worthwhile with Tournament? Play Ironmonger, reveal Province, don’t discard it, draw it, play Tournament, win? Or would you be better off just getting more Tournaments instead of the Ironmongers? Would Ironmonger help you get your early provinces faster than a pure tournament deck?
Is it a viable defense against Ghost Ship?
‘hitting a curse reduces Ironmonger to a straight cantrip’ – Not quite true, since you can discard the Curse.
And since discarding a curse is as good as drawing it in most situations, Ironmonger’s pretty much a lab here too.
Thank you Theory. Great article.
Having been the person who suggested the Ironmonger/Lookout interaction, allow me to explain: as you point out, playing Lookout followed by Ironmonger has no special interaction, since Ironmonger cares about the second card.
But I find that Lookout is usually not a good choice for building trimmed monster engine decks, because you can only really trash about half your copper/estates — after that, the risk of drawing three engine parts becomes quite high, if you’re building the engine efficiently.
Ironmonger helps to solve this by giving you visibility into the next card, so that you know whether you have junk you can trash with Lookout. Not crazy powerful, of course, but combined with Ironmonger’s other effects, possibly good enough to go for an engine requiring strong trashing on a board where Lookout is the strongest available.
I’d take issue with the “strictly better than a laboratory” statement from discarding a VP card with it, as there are a lot of niche situations where you WANT to draw a Victory card. I’m sure I don’t need to list them there.
“Usually better” would be a more precise description, I think.
You can always choose to draw the Victory card.
The first few times I saw Ironmonger I ignored it because I couldn’t figure out how it was useful. I got destroyed in all those games. Just now I played a game without a +action other than the Ironmonger possibility, so I used it. Other cards on the board were Highway, Wharf, Watchtower. My opponent bought two attack cards (Soothsayer, Cutpurse). Just a handful of turns in, I had 7 Highways that I was top-decking as I bought them, and he conceded even though I was down a Curse and two Golds.
Possibly a stupid question: If you Ironmonger and its a Hovel, is it just nothing?