Why is Helmholtz sent away in Brave New World?
Pohnpei397 is spot on, however I think when looking at this question it is also important to consider how Helmholtz reacts to this banishment.
"Whereas, if [Bernard] had the smallest sense, he'd understand that his punishment is really a reward. He's being sent to an island. That's to say, he's being sent to a place where he'll meet the most interesting set of men and women to be found anywhere in the world. All the people who, for one reason or another, have got too self-consciously individual to fit into community-life. All the people who aren't satisfied with orthodoxy, who've got independent ideas of their own. Every one, in a word, who's any one. I almost envy you, Mr. Watson."
Helmholtz rose from his pneumatic chair. "I should like a thoroughly bad climate," he answered. "I believe one would write better if the climate were bad. If there were a lot of wind and storms, for example …"
The Controller nodded his approbation. "I like your spirit, Mr. Watson. I like it very much indeed. As much as I officially disapprove of it." He smiled. "What about the Falkland Islands?"
"Yes, I think that will do," Helmholtz answered. "And now, if you don't mind, I'll go and see how poor Bernard's getting on."
While Bernard completely panics at the the prospect of being banished to an island, Helmholtz is very accepting of the idea. After conversing with Mond about the the type of information he will be exposed to there, Helmholtz realizes that he will be able to think somewhat individually while banished. Furthermore, he will be surrounded by other "radical" thinkers, which he converse with and develop ideas with. While it is unlikely that Helmholtz, or any of these other radical thinkers will be able to follow through on any of their ideas, the simple prospect of being able to think for himself is enticing.
Helmholtz Watson is sent away because he starts to want to have real feelings. Because of this desire, he starts to say stuff in his classes that goes against what people in that society are supposed to think. You can find this in Chapter 12.
I think that Helmholtz is also sent away because of what he does when John the Savage causes the brawl at the hospital by disrupting the soma distribution. When Watson does this, he is explicitly fighting against what the society is trying to do.
So Watson is thinking too much and starting to want the sorts of real feelings and experiences that the society is supposed to prevent.