In Part One, Chapter Five of 1984, why does Winston start to sweat when he discovers that the dark-haired girl has been looking at him?
I think that you can find the answer to this question in the lines immediately following the one that says he is starting to sweat. There, Orwell tells us that Winston is afraid of the girl. He keeps seeing her around and, in that society, he naturally thinks that she is somehow spying on him. In our society, he might think she was interested in him (which she is). But in his society, personal relationships just don't happen. So if someone seems to be watching you, they are probably spying on you for the Party.
You can see this fear of his in the following passage:
Why was she watching him? Why did she keep following him about? Unfortunately he could not remember whether she had already been at the table when he arrived, or had come there afterwards.
If you keep going a bit down from there, you will see that he says that he thinks that she is some amateur spy. He says that they are the most dangerous kind.
When Winston realizes that the dark-haired girl is looking at him, the sweat begins to pour down his "backbone." He also feels a "pang of terror" and an enormous sense of "uneasiness." This is because the girl has been following him around for some time and he thinks that she must be a spy working on behalf of the Thought Police.
At this stage in the story, Winston is already a rebel. He has been keeping a diary, for example, and has started to remember his life before the Party came to power. As such, he is worried that the dark-haired girl will detect his rebellious feelings and report him to the Thought Police. If he is arrested then there is little doubt that he will be "vaporised" for committing Thoughtcrime.
Winston does not yet realize that the dark-haired girl stares at him because she wants to be intimate with him. Until that point, her presence continues to arouse feelings of suspicion and intense fear.