Why does the prime minister "redistribute" Natalie?

In his early scenes with her there's clear banter between them, she's initially nervous and overly-familiar, but he likes it:

David: Hello, Natalie.

Natalie: Hello, David. I mean, sir. Shit, I can't believe I've just said that. And now I've gone and said 'shit'. Twice. I'm so sorry, sir.

David: You could've said 'fuck' and we'd have been in real trouble.

Natalie: Thank you, sir. I had a premonition I was gonna fuck up on my first day. Oh, piss it!

[David smiles at her as she leaves]

After failing to engage her in conversation the first time they're alone together, he not-so-subtly tries to learn some more about her later:

David: Natalie. Erm, l'm starting to feel... uncomfortable about us working so closely every day and me knowing so little about you, it seems elitist and wrong.

They talk about themselves and get to know each other a bit better before he asks his question:

David: Erm, and you live with your husband? Boyfriend? Three illegitimate but charming children?

Natalie: No. I've just split up with my boyfriend, so I'm back with my mum and dad for a while. He said I was getting fat.

David: I beg your pardon?

Natalie: He said no one'd fancy a girl with thighs the size of tree trunks. Not a nice guy, actually, in the end.

David: No. You know, erm... being Prime Minister, I could just have him murdered.

Natalie: Thank you, sir. I'll think about it.

David: Do. The SAS are absolutely charming. Ruthless, trained killers are just a phone call away.

[Natalie leaves]

David: Oh, God. Did you have this kind of problem?

[Turning to portrait of Margaret Thatcher]

David: Yeah, course you did, you saucy minx.

So he's clearly very into her. Then the next time he goes to see the US President, he opens the door and what does he see...

President: Er, Natalie, [smirking] I hope to see much more of you as our countries work toward a better future.

Natalie: Thank you, sir.

Then before either the President or Natalie have time to explain, David makes the speech and asks his secretary to have her transferred out of Downing Street. So really it was just a misunderstanding, David thought she was a willing participant rather than just the sexually harassed secretary as he later realised.

I mostly agree with @NGLN, but I think David is afraid his feelings for Natalie are going to overwhelm him and everyone will see that he's in love with her. We can see that during his speech to the reporters, in which he ends with something about "what is best for ..." pauses for a long time, locks eyes with Natalie, she smiles ... and David finishes his sentence: "for Britain." His love for Natalie, and desire to protect her from the US president, almost overtook him in front of the press on worldwide television.

So David's in love with Natalie, but he's afraid of telling her or anyone that, since he's Prime Minister, but he's also afraid that everyone's going to see how obvious it is that he's in love with her. So the only thing he can think of to do is to move her away.

(Towards the end of the movie, when Natalie gives him the Christmas card that says she's his, he realizes he doesn't care what people think anymore. He's going to go find Natalie and tell her he's in love with her. And he gets a car "right away" and then goes door to door looking for her.)

The prime minister is obviously attracted to her, he expresses (to himself and the audience) the inconvenience of it, and he conveys almost childish behaviour. With and without her nearby.

Before the speech wherein he disavows collaboration with the US, he sees the US president harassing Natalie. That certainly served as inspiration for his speech, although not from Natalie's part, but rather the president's.

The prime minister gives the well received speech, and he got his grip back together. He realizes Natalie's presence is distracting him from doing his job. And to protect himself, he sends her away. Possibly because he might think a serious relationship would not work out with her or would not work within their shared workplace.