It's hard to say, both are common if you ask me. However, "ten pies nas nie lubi" emphasises that this dog doesn't like us (but is friendly towards other people). "Ten pies nas nie lubi" doesn't seem to put so much emphasis as the other one. I don't know if that makes sense, the difference is very subtle.
You wrote both examples the same way ;) In my opinion "nas nie lubi" is overall more common, but when you are contrasting: "This dog does not like us, but likes you." it would be more emphatic to say "Ten pies nie lubi nas, ale was lubi." with additional stress or lengthening of "nas" and/or "was".
I know that there is a difference between word orders and emphasis, but which words are emphasized most? The way the other commentors seem to be trying to say is that the last word is the most emphasized. Is that right? Is there a good website that someone would recommend for farther reading on this. As an native English speaker, put us/nas before the verb is very weird.
I don't know about readings, but yeah, generally the emphasis goes to the end of the sentence.
I#m going to report this: I wrote - This dog dislikes us. Does not like and dislike, it#s the same thing.