Is it illegal for an employer to fire someone who refuses to wear heels if they make her very uncomfortable?

That would depend on some more specifics.

The only argument under US Federal law that would ban firing a person for that reason is claiming a disability under the ADA. It is arguable if that would apply.

That said, I suspect the employee would qualify for unemployment in such a case, and the employer would find it difficult to hire and retain workers.

Not if it is a requirement of a reasonable dress code.  But there was a case in the UK of a company being shamed into changing its dress code by adverse publicity when a woman attracted over 100,000 signatures to her on-line petition after she was sent home by the company for refusing to wear high heels.

Here's the thing to understand about that, if it's req for employment, it's tax deductible. Plus, if a woman twists her ankle and gets hurt because of the requirements, it's called maintaining an unsafe work environment. Big lawsuit. 

No.  Employers can set REASONABLE dress codes.

Call the Department of Labor and ask them to help.

Generally no..... employers are allowed to set dress codes.  As long as you were made aware of the policy before being hired and that policy is being consistently enforced among other employees, the company could justifiably terminate your employment if you refuse to comply for any reason.

It should start with warnings for not wearing the correct attire and dismissal if they continue to ignore it, but yes it's legal, I can't say I'm not wearing my uniform because it's uncomfortable.