Terrence Howard played Rhodey in Iron Man but Don Cheadle played the character in Iron Man 2.
Was there any particular reason why Terrence Howard didn't continue on in the role?
There is no official word on the actual reason for replacing Terrence Howard from the second installment of the movie. Nonetheless, there are some speculative reasons, which were doing rounds in the industry about his departure or better say eviction from Iron Man 2 and the main thing which kept coming was..
According to Entertainment Weekly, Howard was the most paid actor in the first movie, and Marvel was finding it hard to renegotiate the deal with him for Iron Man 2.
“Howard was the first actor signed to the film and, on top of that, was the highest-paid. That’s right: more than Gwyneth Paltrow. More than Jeff Bridges. More than Robert Downey Jr. And once the project fully came together, it was too late to renegotiate his deal.”
It was also reported that, director Jon Favreau was not happy with Howard's work. Quoting from the same article, referenced above
Favreau and his producers were ultimately unhappy with Howard's performance, and spent a lot of time cutting and reshooting his scenes
The unhappiness with Howard's work led Favreau and his screenwriter to trim down the role of Howard's character in the next installment.
As such, when Favreau and screenwriter Justin Theroux went to map out the sequel they found themselves minimizing Howard's story line. Once Marvel learned that Favreau was thinking of curtailing the role, the studio went to the actor's agents with a new and drastically reduced offer — a number that's similar to what supporting cast members were paid for the first movie. The agents, according to sources, were so taken aback by this new figure — estimated at somewhere between a 50 and 80 percent pay cut — that they questioned it
Marvel never commented on the above speculations officially.
It's due to huge pay cut as rumor suggests and even due to role cut as indicated in @Incognito's answer.
Here are Howard's reaction on the removal from Iron Man 2-
"Marvel made a choice, and it was a very, very bad choice. They didn't keep their word. They didn't honor my contract," Howard tells Parade magazine. "They produced a great bounty with the first one but they put it all in the storehouse and you were not allowed in.
"They did the same thing with Gwyneth Paltrow, from what I've been told. They did with almost everyone except Robert Downey. One of the things that actors need to learn to do is always stick together, one for all and all for one."
"I've seen the Iron Man 2 script, I know what's going to happen, but I'm not revealing anything. I believe in karma. When someone does something wrong, you don't have to get them back. Everything right will return the favor for you."
"I'm definitely looking forward to seeing the movie," says Howard. "I want to see Don Cheadle become me. No, I want him to do better than me. That's what I really want to see. Don Cheadle was the reason I got Crash, so I have a lot of love for him." (source)