If you resign for good cause attributable to the employer you can get benefits. I don't know if this would qualify.
If you are terminated without good cause, you can get benefits. Again, I don't know if this would qualify. I imagine a lot would depend on whether or not your fear of dogs could be classified as a disability and if so if your employer made a reasonable attempt to accommodate that. An employer is permitted to have a dog-friendly work place. The productivity issue would be arguable - people are still entitled to unemployment benefits if they are unable to perform the work. So you might be considered unable to perform the work.
So as far as unemployment, my thought is it would be best to be terminated. However, you would also then have to explain that when you looked for a new job. Only you can make the ultimate decision, and either decision carries risks.
You may want to consult with an employment law attorney regarding any possible issues as to disability protections, and for current case law on the unemployment issues that I have described.
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