Remember that you don't have to be a Dragon race, it's that you have the Dragon type. There are several creatures with the Dragon type that aren't exactly Dragons.
Anyway, the rules have this to say:
Prestige classes allow characters to become truly exceptional, gaining powers beyond the ken of their peers. Unlike the core classes, characters must meet specific requirements before they can take their first level of a prestige class. If a character does not meet the requirements for a prestige class before gaining any benefits of that level, that character cannot take that prestige class. Characters that take levels in prestige classes do not gain any favored class bonuses for those levels.
As far as the RAW is concerned, you must meet the requirements of the Dragon Disciple class in order to take the first level in it. This means that, once you fulfill the requirements, and then proceed to no longer fulfill the requirements, you could still take levels in it, per RAW.
Per RAI, I'd argue that the intent is that you should qualify for future prestige class levels as long as you meet the pre-requisites. In other words, if you meet the requirements in one level, and then don't in another, you could take the prestige class levels at the levels that you qualify, and then can't for the levels that you don't qualify.