Woah, woah, woah. WHAT? I'm out. I'm still traumatised by the horse in Shadow of the Colossus. I don't think I can take a dog-based loss. Especially when we've established Snake's wolf is adorable.
The latest news out of TGS (via NeoGAF, and a Dengeki interview translated by @) is that the dog, called DD (for Diamond Dog), Quiet the Sniper and even Snake's horse, can all suffer permanent in-game deaths. The three form part of a buddy system that lets you pick companions to come with you on missions but all risk being killed if you take them with you.
From the Dengeki interview: Q: If a buddy takes too much damage, do they die? Kojima: Yes. — JunkerHQ.net (@JunkerHQNet) September 22, 2014
Q: You mean they're dead for the rest of that mission? And they return to Mother Base wounded? Kojima: No, they're dead, period. — JunkerHQ.net (@JunkerHQNet) September 22, 2014
Q: Really? They die for real?! So in that stage demo, if Quiet had died, she'd be gone from the rest of the game? Kojima: That's correct. — JunkerHQ.net (@JunkerHQNet) September 22, 2014
Only Quiet's abilities have been outlined so far. She can be ordered to scout ahead or positioned to provide covering fire. The horse is pretty self explanatory but it's not clear yet whether the dog can sniff out supplies, attack or do anything like that. It looks like there are several extra characters in the game to be discovered as you can bring multiple companions - the chopper you use to reach missions has four spare seats available.
These parters will come with their own side missions but their (potential) deaths will have no impact on the core plot - it's possible to not even find them or leave them out completely and finish the game without them. We know how you find the dog though:
Honestly, he is never leaving the base once I get him home.