Yes. There's a site in Denmark that provides this service. You can try it for free for, I think, 20 SMS.
The number is +45 40606361
If you send an SMS to that, with
EGLL as the content, you will get METAR, TAF, long TAF, and a bit more for EGLL. Each of those counts as one SMS, so you get about two goes at EGLL and your 20 are gone.
It knows most airport ID codes
You can limit what it sends by the use of more codes. For details, go to
www.wxsupport.com
Then you pay! I've got a sub there, which I use when abroad (since the iPAQ and Bluetooth to the Internet is horrendously expensive outside the UK).
It does have occasional glitches - last time I was in France and needed it in anger, it was sending out-of-date information. Since then, when I've used it just to make sure, it's been fine.