No.
There are pictures of them holding hands, yes, but it's for publicity. Both have come out with recent albums, and a 'relationship' in the news is sure to bring more attention to them.
More then once, Harry has gone to Taylor's hotel and snuck out shortly after, only to sneak back in the morning. Why? So the paparazzi could get pictures of him 'leaving' the hotel after 'staying' for the night.
Also, I have heard that there is one occasion where Taylor grabbed Harry's hand right when the paparazzi showed up, only for Harry to drop he hand shortly after the pictures were taken.
Believe what you want, but I think this is bullshit.
EDIT:
Just adding more evidence, you know.
The video, Louis says: I'm happy they're
good friends.
If you listen closely at :27, just after that comment, he mumbles "If that, even." This suggests that they're not even friends.
Maybe, just people helping to publicize each other.
link the boys were asked who was single. Look at whose hand is in the air. I'm not talking about Niall's.