I think depends mostly on how you approach it.
Really, the interviewer doesn't even have to know that you're a furry, but if you must tell, then ultimately, your attitude determines the outcome.
If you do tell the interviewer that you're a furry, his or her reaction will be determined mostly by how you present it. If you're shoddily dressed and wearing dog ears and a tail, and when asked about it, you just say, "I'm a furry and you'll just have to accept that," you probably won't get a good response. You'll come across as having a bad attitude and not taking the interview or the job seriously.
If, on the other hand, you're cleaned and trimmed, neatly dressed in a suit and tie, and yet still wearing your dog ears and tail, you will most likely peak the interviewer's curiosity. You're obviously taking the interview seriously, so there must be reason for the ears and tail. It's pretty safe to say you can predict the interviewer's first question and have an answer ready. You could say something like, "The tail reminds me to wag it instead of my tongue, and the ears remind me to listen before speaking." From there, segue into what it means to be a furry and how the experience makes you the right person for the job. In this case, you have set yourself apart from the crowd in a positive way, and in many cases, will increase your chances of getting the job.
That won't always get a positive response from the interviewer, but if it's done creatively and with a good attitude, it will usually improve your chances. While some people will have a negative response due to some sort of prejudice, most will not. Anyway, do you really want to work for the person who is prejudiced? Bear in mind that a company may get hundreds of applications for a single job opening, so while most people want to blend in, standing out from the crowd ultimately helps you get noticed and get the job.
Personally, I finally started seeing success at work, and in life in general, when I finally decided to quit listening to other people and instead be true to myself, which includes being a furry.
So in the end, I think it's you're attitude that determines your success.