Your suspicion about formaldehyde in no-iron shirts, also known as wrinkle-free, is spot on.
To be sure, you could write to the company and see if they disclose the chemical treatment or hide behind proprietary rights.
I hate to iron as much as the next person, but to make cotton iron-free, the fabric is typically soaked in formaldehyde. When you press it flat it becomes wrinkle-resistant. Retailers warn that due to the change in molecular structure of the cotton shirt, the fabric becomes more brittle. This means the life expectancy of your husband's no-iron shirt may be cut by 25 per cent.
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If a short lifespan isn't discouragement enough (you want to stretch every dollar, after all), these low-maintenance duds are impregnated with a cancer-causing chemical.
Since formaldehyde is known to emit fumes from products and can be inhaled—most of the cancer research on formaldehyde has focused on risks from inhalation—avoid it. Off-gassing of formaldehyde from building products, for example, is a concern for indoor air quality. Health Canada recommends the reduction or elimination of as many sources of formaldehyde as possible.