Use with care. Antiperspirants are astringents that chemically pucker up the opening to the sebaceous glands in the armpit. However, they do not stop the secretion of sebum, which build up in the glands while the antiperspirant works.
If the sebum is permanently blocked over long periods of time, infected cysts may develop. Eventually, in rare situations, this can permanently damage the duct and then the only solution is to surgically remove the painful lumps that develop via an I&D procedure.
It's all about common sense and discretion, it is a temporary effect but using it too often like applying it everyday may cause some harm though not recorded or reported. You may try surgical approaches to sweaty palms too.
Just to add on antiperspirants do not plug sebaceous glands but sweat glands and sebaceous glands are only the most active around the T zone (face) not so much on palms where the cornified layer is the thickest.
ps: I do not sell or represent any of these products ;p