I have NOT run Dungeon World online, but I have run two other *World games online, and I've run Dungeon World at a table. I do have a lot of experience running online, and the playstyle I've evolved is similar in a lot of respects to what Dungeon World is trying for.
Dungeon World's biggest challenge, regardless of playing venue, is the lack of structured order. If a system has a turn structure and rules for when you can make your declaration, the only challenge online is waiting for your turn. But Dungeon World relies on a lot of fast, often chaotic back and forth. This presents some issues in a text-based game. You need a few tweaks to account for things like varying typing speeds, and the fact that typing in chat isn't quite real time like talking is.
In my opinion, the relativity of speed between voice and text is absolute enough that anything will translate provided the players don't mind the slower pace, and can use a few tricks to help the text flow more easily. As with the rules of the game itself, make sure players know and agree to any meta rules you choose to use, such as those suggested below.
If the client you are using has this option, use it. It's darned useful to know when someone else is typing something. And when no-one is typing anything.
Keep it Short
A GM can and often should use long blocks for descriptions or perhaps an "off camera" narrative ("Meanwhile, back at the farm..."). But during any scene, or any free play, the GM and players both should keep their poses in easily digested bite-sized chunks. No more than a sentence or two at a time.
Why? Two reasons, one obvious, one perhaps less obvious. First, typing a huge block takes longer, which means people are sitting around waiting longer and/or getting their say in first because you took so damn long. :)
But second, a long, unbroken block of text often feels like it can't be interrupted once it's posed. It interferes with the back and forth when a player feels like he can't react to the first line in a ten line opus.
A corollary to this: Players, don't be afraid to challenge something just because it was part of a huge block text! Don't be annoying about it, but you should always feel okay about simply saying "I want to interrupt that action. May I?"
GMs, when you DO need to type something long, tell players. ("Long Description Coming..."). Better yet, have it typed and ready to paste in. If I have to do a long thing but haven't been able to prepare it (Improv, FTW!), I still try to break it into parts and use an ellipsis or "(more)" at the end of each part to let players know more is coming.
Ignore Typos and Spelling
This is not just an issue of being annoying. It has a practical application. If you make someone self-conscious about their typing, they will type slower. This is an important consideration all the time, but in Dungeon World, the play depends more on the rapid-fire back and forth. Make it a rule that no-one cares about your typing. Make sure players are comfortable knowing they can dash out their sentences without fear of being nitpicked.
No Dead Air
Often in the text environment, you'll find dead air as everyone thinks someone else should be replying.
As a GM, don't allow dead air. If a minute passes and no-one has typed anything, move on. Make ninjas attack, or a horse throws a shoe, or Zeus comes down to seduce someone. Or just cut to the next scene. Something. Keep it flowing.
As a player, don't create dead air. If you have something to say, say it. If you have nothing to say, say that. A quick IC or OOC cue to let the others know you didn't just wander into the next room.
As player or GM, if you think you're done with a scene, let people know you're ready to move on. Don't necessarily force others to move on, but let them know you have no more to contribute to this scene.
As a player or GM, always let the group know when you need to be away from your keyboard for more than a few seconds. "BRB Guys." A name shift to AFKNickname. Something. Don't just wander away and leave everyone guessing.
As GM, go out of your way to provide chances for players to interrupt you. This is more than just the bite sized blocks of text. ASK and OFFER these chances.
Sometimes I like to provide a countdown. I will set up an action, then count down. Players have until the count reaches zero to post something. Even something as simple as "Wait!" is enough. If they do, I let them pose their interrupt. In no-one can even be bothered to type "wait" before I finish, then something happens to them. :)
Ensure Everyone Gets a Turn
This is going to go a little bit against the Dungeon World philosophy. But... many people type at vastly different speeds. If all your players are fast, ignore this. But if they aren't, an adjustment needs to be made.
In a conflict or any space where timing matters, the GM must enforce the order of outcomes, which will often not match the order of presentation. ASK the slower-typing player "Bob, you have something to say?"
Don't punish the slower typist for being slower.
In a lot of play, action goes something like "I swing my sword." "Okay, roll to hit." rolls dice "I hit." "Okay, roll damage." rolls dice "I do 6 damage."
Dungeon World encourages a bit more proactive rolling. As a GM of an online game, I recommend you require it. It saves a lot of time. If the roll wasn't necessary, it still overall saves more time to have the player just roll as he declares every time. Even if the GM has to interrupt for a reason, or apply a modifier, it's faster to have that roll out there.
Players, learn the dice notation commands for your client! Nothing slows things down more than constantly asking what the command is to roll 2d6!
I've used all of these tricks, and they all work. They do all require players to buy into it. I've had players who just couldn't resist nitpicking other people's typos. Or who would always type long, beautifully detailed descriptions of their actions. It. Slows. Play. Get your players to buy into these tricks, and your game will flow faster. It still won't be as fast-paced as a table-top, but you will see a noticeable improvement in speed of play.