How do I stop my dogs from cleaning the cat box?

Go to the pet store and take a look at a new litter box. There are models that have curved ramps and covers that conceal the "entrance" to the box. There are models that require the cat to jump in from the top, and is supposed to keep the dogs from being able to get to the litter. It may be worth a try to invest in something like that since you can't reduce access any other way.

Can you place the dog in a different room from the litter box?

You could put a baby gate at the door of the room where the litter box is that way the dog can't get to it but the cat can. Good luck!

Persistent dog training. However, this is hard because cat food is very rich. In turn, to a dog cat feces is a rich treat. The only alternative I can think of is getting an automatic cleaning litter box. The only problem with that your dog, in some cases cats, like to play with it.

My parents had that problem. What they did, is get a table, and put the litterboxes on the table. The cats quickly figure out where the box is, and have no problem hopping up to get to it. (Now, with an older cat, that could not be a possibility, if they have joint problems). The dogs were unable to reach into the litterbox that way.

P.S. We also used hooded litterboxes, for then the dogs couldn't just tilt it to the side.

we put a baby gate up in the doorway of the room the litterbox is in. It keeps the dogs out and gives the cats something else to climb on.

We have a pekingnese that think it (cat poop) is an food. So I put the litter box up high enought that the cat can jump to it and the dog cannot reachi it.