I agree with Mr. Chambers above. In addition, note that the example in the original question is actually a bit of a set phrase: "it won't/doesn't hurt" e.g., "A little pressure won't hurt" "Give it a try, it doesn't hurt to learn something new" used when encouraging someone to take a small risk or try something new or different. You never use "harm" in this context.

In my opinion hurt means the mental torture and harm belongs to physical pain.

If you harm someone or something, you cause damage. If you hurt someone or an animal, you cause pain. So hurt is a specific form of harm (and pain is a specific form of damage). If you are late for a job interview, you harm (or damage) your chances of getting the job. We don't normally use "hurt" here in everyday English (it sounds a little literary) but I couldn't say it was wrong. If you cut your finger when chopping carrots, you hurt yourself. You could also say "harm" yourself but this would be less natural and less precise.