The obvious choice for an MMO on a handheld device is a text or browser based game. A browser based game rarely requires any quick interactions, which suits the lack of controls, and they don’t require much to run smoothly. As a bonus, most are also free to play.
You can find a browser game for pretty much anything. From running a farm to running a kingdom, from levelling a single avatar to levelling an entire army. Social networking sites, such as Facebook, have plenty of them. You can find them advertised on pretty much any gaming website these days, including here on Bright Hub.
Handheld MMO games are not solely trapped in browsers. Several MMO clients have been developed for the Blackberry and iPhone, which are worth checking out.
We have the EVE online rip-off, Outer Empires for the iPhone. If you prefer a fantasy MMO the iPhone also has Hybrid: Eternal Whisper. For the Blackberry, we have the fantasy Path of the Warrior. If you’re looking for one to play on both, check out tibiaME.

click to enlarge
It's not a complete list, there are other games out there, I’ve had a look at some of these, they won’t blow you away but they are playable. If you don’t want to settle for a browser game, they’re worth checking out.
It is worth mentioning that games like EVE Online and World of Warcraft haven’t forgotten about handhelds. Don’t get your hopes up; they don’t have game clients just yet.
Both the Blackberry and the iPhone have an EVE application which lets you trade and keep an eye on your characters trading. WoW has an iPhone app which gives you the armory without using the browser. The Blackberry has a third party version but it’s very poor, sticking with the browser is your best option.

click to enlarge
If you really want to play your normal MMO from your handheld device, there is one way. You can leave your desktop and remote control it from your handheld. Technically, you will be playing an MMO with your handheld device. In actuality you will be looking at a 3 frames per second slide show. You’ll be able to connect, possibly chat with friends but not much else.
Just keep in mind, before you try any of these, make sure you have a good data plan or deep pockets. You could run up quite a phone bill playing one of these.