An undershirt is just that, an undergarment, and should be worn as such. It is not a fashion statement.
The whole point of an undershirt is to minimize the amount of body fluids that stay on your skin. It is not just sweat, but slight amounts of body oil as well. A soft cotton undershirt will absorb and trap these better than a dress shirt will. Sweat also contains bacteria and Urea (which is in urine). Once an undershirt stain cannot be removed by cleaning, it should be thrown out.
The chemicals in antiperspirants also contribute to staining, and make the stain more difficult to get out.
And, keep in mind that when you use antiperspirants you are only using them under your arms where you sweat the most, but your entire body has sweat and oil glands. Think of the guy with the soaking shirt after a game of pick-up basketball, he didn’t just sweat from his pits, and neither do you – even when you are just moving around the office (just a lot less). And, stress or motion can cause you to sweat, Winter or Summer.
Soft cotton is good at absorbing sweat and oil (wool is even better which is why it makes a great undergarment for winter sports). Poly blends are terrible at absorbing anything and a lot of the mid to lower end no-iron dress shirts are poly blends. Check your dress shirts and see what the fabric is made of.
The issue isn’t really about staining your shirt, it’s about the bacteria, urea, and oils that are not wicked off of your skin properly and remain there to compound and block the pores of your skin. It won’t kill you (usually), but why not wear the undergarment for its intended purpose, and keep your body a little cleaner?
