What happens if I don't give my dog water?

It seems that this answer will depend on what version of Stardew Valley you are using, as there were changes to the benefits of filling the water bowl with version 1.4 of the game.

Pre-1.4:

The water bowl seems to be only cosmetic. If you don't fill the bowl with water, there seems to be no ill effects.

Per the [Stardew Valley Wiki] (at the time) (http://stardewvalleywiki.com/Animals#Pets), in the 'Pets' section:

There's a bowl on some tiles on the top left side of your house, you can fill it with water for them. The bowl is purely cosmetic.

There's also some other links that show users that did not give their pet water (some due to not knowing you could) without any repercussions.

Versions 1.4 and up: The bowl was updated to provide a small amount of friendship when filled from empty.

Per the updated Stardew Valley Wiki, in the 'Pets' section:

There is a bowl on some wooden tiles on the top left side of your house that you can fill with water for your pet. If you water your pet's bowl, the next morning your pet's friendship will increase by 6 points. (Note that this event will not trigger if it is raining the next morning.) On rainy days, your pet's bowl will automatically be full. If it rains before adopting a pet, the bowl will remain full until adopting a pet.