That's very sad. I've just lost my dog and it's one of the most painful things in the world, she meant far more to me than anyone outside my close family. Hopefully the vet may be able to do something or he/she may get better on his/her own - whilst sometimes the very difficult decision to have a pet put to sleep has to be taken because it's not fair on the animal, sometimes, vets are too quick to suggest this option and owners too quick to agree. I was advised to have my rabbit put down numerous times, but because he was a real little trooper and we were so devoted to him he got better and lived for another 2 and a half years! So it's just something for your boyfriend to bear in mind, depending on what the situation is, he/she may get better!
I would say just be there for him, let him know that if he wants to talk you're all ears, and perhaps share experiences about your own pets, past and present. He may not want to talk about it all, or he may want to talk about it a lot, you will be able to gauge from how much he says. If he wants to talk, talking about happy times with his dog may help. Try to be positive and don't assume too much until you know exactly what the dog's condition is.
I hope his dog gets better!