Can your command make you clean someone’s house?

Not only but no F'ing way. I would take an article 15 first.

Force? Most likely not. Get some "Volunteers" together, more likely. If I had a buddy that was having issues like this and everyone was gone, then by all means pitch in. The last thing they need to do is come back to a house that is unlivable and that has more than likely attracted bugs or vermin. Clean it to a point. Doing a bit of laundry helps, but you don't need to fold it and put it away. Wash the dishes and clean the kitchen. Ensure the Bathroom is sanitary. Close off areas that the dog may be going to that it should not. Bear in mind, they do not want to feel violated when they return.

When you joined the military your option of choosing to make decisions are over. Failure to follow a lawful order can get you sent to the brig.

I suggest you analyze this a little deeper. Imagine yourself in the boots of the other troop...you come home from deployment and find your usually-nasty house is spic-n-span, and you know your wife didn't do it because she's gone, too. While that is sifting through your brain, you come back to the unit and find out that a half-dozen of your "buddies" went in there and gave it a GI party because it was so disgusting. Think of the embarrassment! Sometimes you have to create a significant and memorable emotional "event" to change people--hopefully for the better--let's hope this does.

Can they force you. As someone once said maybe they can’t force you but they can make you sorry you didn’t comply. Let’s take three scenarios. You say no and they bring you up on a charges and now you before somebody about the UC MJ. My guess is that you’ll win is it unlawful order. Scenario number two you tell your NCO you don’t wanna go and he says something like F#@#$ You and then tells the rest of The people go do the cleanup after that you’re ostracized no one will work with you no one will talk to you no one trust you. Did you win? Third scenario you tell the NCO no and then for the next month or more you seem to be getting a lot of extra details and a lot of extra special duties and your efficiency report at the end of the year doesn’t look that good. This one has two questions did you win? Can the NCO do that?
So here’s my point you signed up you said you’d be part of the team and here’s a test of that commitment. A lawful order Be in dress uniform and ready for inspection at 0800 tomorrow. An unlawful order get your gun and go downtown and rob a bank and bring me some money. In between there’s a lot of gray area and no one can tell you for sure what’s right and wrong but I can tell you that in most cases you should give your commander the Benefit of the doubt and pride yourself on being a team member.

It's probably more we're a team and gonna help out teammate out rather than order. Either way unless your not planning on retiring you'd be better off doing it. It's kind of like cutting grass with scissors not right, but we do it

What about the family's right to privacy? I understand, and agree, with wanting our troops to live in a clean house, but a civilian also lives there. I disagree with going into the soldier's home with a bunch of people to clean and invade his/her privacy. I know how bad things can get because I grew up with a slob of a mother. After my father died when I was a teenager three of us kids took over the cleaning, the fourth did not. When us three left and the fourth stayed behind it became a hoarder house. I have to disagree with taking other people into a house to clean it without one permission and without one of the family members there.

This is a tough one. If I were the HMFIC I would contact the Housing folks and seek guidance from them. As most housing is privatized, this is probably not within the scope of his command's authority.
Now, if his buddies gave a damn about him they would have already fixed it without involving his command.
Hind sight is 20/20 and there are likely details that are missing to this saga. Repeat offender???
So, if the order still stands, ask your buddy if he is willing to withstand the shit storm of refusing to comply. If so, he must be willing to be held accountable. It is not unethical to expect SMs to help a teammate out when they are in need.
Or just go clean up the dog shit and drink all of his liquor and beer.

I think other posters covered the order aspect. Now let’s look at alternatives to depending on the unit to watch a dog. There are people that volunteer to look after pets of service members. Recommend this person is counseled to pursue an alternative that is a more viable care plan.