Having been a devoted wearer of only New Balance shoes for nearly 20 years I can tell you the problem isn't a result of cats peeling in the shoes as some have said. I first noticed and reported the issue to New Balance way back in 2007 but looking at some of the earlier comments here, the issue goes back to 2005 for some as well. I too was quite puzzled back then what was causing my 1122 running shoes to produce a cat-urine like odor seeing that I don't own any pets and keep my shoes indoors.
The problem was never addressed, although I will say New Balance has some of the best customer service around. The problem is, they are the front line and have to field all of the complaints from people like us, but do so in a very professional manner. This problem then continued on with the 1123 model as well, and I have also noticed the issue in the 856 cross trainer. There are other models affected that I have read online, but don't personally wear.
The problem is the adhesives/glues that New Balance was/is using in their shoes. This material would break down when coming into contact with water, and then allowing water to get into the shoe and thus causing mildew, which is what is producing the cat urine-like odor. While New Balance continues to excel in customer service, they also continue to take a downward spiral in overall quality. Personally I think it is because of lesser grade material cutting production costs in my opinion, but it is clear that they have acknowledged the problem and take care of those who finally figure out to contact customer service, rather than toss the shoes away. This is a very large issue, and doing a recall of all models affected would cost the company several millions of dollars, which is why they gladly will take care of the customers who make the effort to contact THEM directly.
Some comments from other people online say to put your shoes in the washing machine...that is a MAJOR mistake and highly NOT recommended. These people seem to think it is something else causing the odor, but they clearly don't realize it is a water/moisture issue breaking down the glues, and in a washing machine, you are basically submerging them in....water which will cause the issue to intensify.
If you are experiencing cat urine-like odors, it's not something you have done, it is the materials in the shoe breaking down after coming into contact with moisture (sweat/water). Contact New Balance customer service, and they will take care of you. I have noticed that replacement of the same model shoe has not fixed the problem. The 1122 which had the issue was replaced with the 1123 which also had the issue. That model was replaced in late 2011 with the 1540 and so far I have not noticed the issue in that model shoe, although have noticed another issue unrelated in the two pairs of it I have owned.
My best advice, after contacting New Balance, at first try the same model shoe that you have noticed the problem in, and if the problem continues, then try a different model altogether. New Balance is beginning to re-number and redesign many of their shoes, and hopefully this issue will be a thing of the past soon, after 5 years, one would think it would have been resolved, but so far it hasn't completely.
I still support New Balance to this day, while this issue is extremely irritating, I have confidence it will be corrected soon, but overall they are my shoe of choice, and their customer service personal can't be beat.