There's a picture of a celebrity using my product on a random instagram account that I was made aware of, so I reposted it on my social media. Unfortunately there's a dog in the same photo (third party, not the celebrity's) and of all things, the dog owner has reached out and is allegedly upset that i'm using the photo with their dog in it. If I would have even thought about something as ridiculous as this happening, I'd have cropped the dog out of the pic in the first place. I don't even know where to begin on how to respond to the so-called offended person. My question is, do I take down the picture? Is there anything to actually be worried about if someone's pet animal happens to be in a picture?
What is the name of your state? If not in the US, what is the name of your country?
Did you get permission from the copyright holder to publish his/her photo of the celebrity on your social media page? If not, you have infringed on the copyright holder's rights and you should remove the picture before it is removed for you.
Unless the dog is a famous one that is a company trademark (e.g., the Taco Bell chihuahua), the dog and its owner do not have rights to enforce.
I would worry about the photographer/rights holder, however. The publication of that photo potentially could get you sued.
If the celebrity has been pictured using your product and you wish to capitalize on this, you should contact the celebrity and/or the celebrity's agent to see if the celebrity would like to endorse your product.
You can respond to the dog owner how you wish. For the celebrity picture, you might want a personal review by an IP attorney in your area. The attorney consultation becomes more important if you are sued for infringement over its use.