Why is what happens to Syme important? (chapter 5, book 2)
At the start of Book 2, Chapter 5, Syme basically just disappears. Winston notices that it is almost as if Syme had never existed. To find why this is important, I think you need to look at Book 1, Chapter 5 where Winston thinks about Syme a lot.
Basically, Winston knows back then that Syme is going to get killed. He says that this is because Syme thinks too much. He is a really good Party member -- believes all the right things. But he is too much of a thinker. Here is a quote showing this:
Yet a faint air of disreputability always clung to him. He said things that would have been better unsaid, he had read too many books, he frequented the Chestnut Tree Café, haunt of painters and musicians. There was no law, not even an unwritten law, against frequenting the Chestnut Tree Café, yet the place was somehow ill-omened.
Syme's death, then, is important because it shows that you don't really have to do anything wrong for the Party to see you as a threat. If you just seem like you have any independent thoughts or any desire to think, you are a danger.
Winston admits that Syme is quite intelligent, but he also assumes Syme may be too intelligent, and he predicts early on that Syme will be "vaporized" by the Party. But Syme is a pretty loyal Party member, so why would they vaporize him? How is he a threat? How could he be too intelligent? A massive part of the Party's power is keeping the people ignorant, or at least stupid enough to keep them under control, and Syme is a possible threat to that. He reads a lot, he thinks a lot, and he knows a lot. Even though he is a loyal Party member, he still thinks too much. Syme's disappearance serves as a prime example of how even your thoughts or your intellectual capacity could make you a target and that the Party can and will "vaporize" you just for showing potential of being a threat, no matter how loyal you may be.