4E D&D is all about the cool combat powers, most of which really require a combat grid to use effectively. Lots of things like "on a successful hit, slide your opponent one square in any direction". Much of the joy of playing 4E comes about by seeing the combination of the powers in your party, having interesting combos, interacting with the environment in cool fight scenes, etc. It's a great game for what it does, which is largely tactical-level combat.
But even around a table with everyone knowing the rules, a single combat can take an hour or more. It would seem to me that for forum-based play it would take much, much longer if you want to keep the same level of granularity. I'm not saying that it couldn't be done - it clearly has been in the past. But you probably have a choice between ditching much of the 4E system, just focusing on the classes/races/setting, or sticking with the nitty-gritty and being resigned to having really drawn out sessions.
I think it's worth asking why you want to do a 4E D&D game in particular? What does using that ruleset bring to the table, so to speak? Is it just because D&D is popular, and so it might be easier to get players? There are tons of roleplaying systems out there, ranging from the very simple to the very complex. It might be worth exploring other options. I'm pretty sure there are systems that are designed from the ground up to be used for things like forum-based play.
If you do decide to stick with 4E D&D, one off-the-wall suggestion might be to ditch most of the rules, including all of the combat rules, and just focus on skill challenges. The skill challenge system in the rules is fairly lightweight and elegant. It could be used even for combats. For example, in this fight with the goblins the party needs to make 6 successes before getting 3 failures, or else the goblins end up winning and taking the party captive, etc. Then players can roleplay using their skills - including their combat skills - to garner the successes they need. Just a suggestion, but I know of tabletop groups that have done this and have been happy at the speedier games that result.
Best of luck, no matter which way you go.