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.
It was also reported that, director Jon Favreau was not happy with Howard's work. Quoting from the same article, referenced above
The unhappiness with Howard's work led Favreau and his screenwriter to trim down the role of Howard's character in the next installment.
Marvel never commented on the above speculations officially.
Its due to Huge pay cut as rumor suggest and even due to role cut as indicated in @Incognito's answer.
Here are Howard's reaction on the removal from Iron Man 2-