You'll have trouble maintaining sufficient chlorine in the pool without it (at least 1 ppm per Ben's best Guess Table). But, if your're willing to test the water for chlorine at least daily and, especially before anyone goes swimming, you can probably get away with it. It'll be much more work, though and you'll probably go through a bunch more swimsuits and expose yourself, you family members, and your dog to much higher levels of disinfecting chlorine than you would with some CYA in the water. You will also, of course, use much more bleach.
You might want to consider a low CYA pool; something around 20 ppm will give you significant benefit without exposing your dog to much potential harm. This post may be instructive in that regard.
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