Stressing each part of that sentence separately, I got five different meanings. English is a strange language.
WHY is that dog looking at me? - the standard version
Why IS that dog looking at me? - a reply to the above; I'm wondering the same thing!
Why is THAT dog looking at me? - why not the other dog?
Why is that DOG looking at me? - why not the cat?
Why is that dog LOOKING at me? - why is it just looking, not sniffing etc?
Why is that dog looking AT me? - why not look past me at the food on the table? (okay, this one's a stretch)
Why is that dog looking at ME? - why not you?