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.