Updated As of May 2014, it works in both directions now! As this edit demonstrates, from somewhere under the channel :)
Yes, you can! But currently until very recently, it was only in one direction...
As detailed in this IET article from July 2012, Eurotunnel (who own the tunnel) have fitted leaky feeder aerials, base stations and repeaters to both tunnels, and it only took 10 months to kit out. Eurotunnel are then leasing them out to mobile operators.
The South Tunnel, which is normally used for trains travelling from France to England, has been assigned as "France" and leased to French mobile phone operators. Service in this tunnel went live in July 2012, and you can now use your phone to call / text / surf whilst travelling from France to England. It's using French mobile operators though, so UK phones will be charged roaming just as they do in France.
The North Tunnel, which is normally used for trains going from England to France, has been designated "England" as is to be leased to English mobile phone operators. However, despite being ready in 2012, it took until May 2014 for the UK operators to sort themselves out, and as such mobile signal has only just started working in the UK to France direction. (Quite why the UK networks were so useless is a different question...)
The service from France -> UK, works well though, I can personally confirm it from earlier! If you're on the phone as you enter the tunnel, your call will continue uninterrupted, and work fine for the whole tunnel. However, just before leaving the tunnel your call will drop, as that's where the French network ends. A few tens of seconds later your phone will sign-on with the UK network, and you'll be able to call again.
The service the other way, from the UK to France, still (May 2014) seems to be suffering some teething problems. Mobiles do work, with the odd drop-out, but data is pretty flaky and suffers really slow speeds + long ping times. It's possible they'll sort that later, but for now internet largely only works in the "French tunnel". (I have managed to post to Travel.SE from the English tunnel, but it took most of the trip for the post to go through!)
AFAICT, there is no mobile service offered in the third tunnel, the service tunnel, only the emergency radios work there.