Kill grass but not dog

Are you trying to kill JUST the grass and no other plants, or is this an area where you don't want ANY plants growing? If the latter, you have more options. Boiling water is effective immediately, and I've heard good things about using salt, assuming the species of grass isn't salt tolerant (probably not an issue unless you live close to the ocean). Salt's effects can linger though, so be really sure you don't want anything growing there if you use it. Also salt can irritate pawpads if your dog walks on it frequently, so you still may want to keep your pup away from the area at first, but it's not toxic. I have heard of people using bleach solutions, and bleach does eventually break down into salt and water, but I haven't tried this myself nor do I know what the right bleach to water ratio is, how long it takes to break down once used, etc so do your own research. If you go this route, please please please do not mix bleach with anything other than water, no matter what anything on the internet says - dish soap, vinegar, and of course other cleaners like ammonia can all create dangerous fumes when combined with bleach. Not safe for you, let alone the dog!

It'll be a lot harder if you want to kill the grass, but not anything else growing in the area. Don't have any suggestions for that, I'm afraid, aside from manual weeding and possibly using a good mulch, groundcover, or weed barrier in spaces around the plants you want to keep alive. Most weed killers aren't very selective.