Why does Winston start to sweat when she discovers that the dark-haired girl sitting next to him has been looking at him in 1984?1984 Chapter 5 Book 1

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:

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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.