Apparently, it’s not the anti-perspirant, so much as what happens to your anti-perspirant in the dryer. Your same question was answered by a dry-cleaner in a GQ “style questions” type column a couple years ago. They said that the heat of the dryer leads to a reaction with the alumnimum in the anti-perspirant that causes the staining.
I tested out their reasoning on two white shirts, drying one in the dryer and hanging the other on a line for a coule weeks, and the line dried one didn’t get anything resembling a stain.
with that said, I’ve been using arm and hammer antiperspirant lately, and haven’t noticed much staining at all (and i’m drying shirts in the dryer).
Than hang my shirts to dry I will! Thanks.