Batman does wear armor. His costume over the continuities has always been some variation of Kevlar, Nomex or other bullet-proof, fireproof, frost resistant, cloth or light plate armor worn under his costume or woven into the costume itself.
He has pointed out that the bat symbol on his chest is to make his chest an easier and more inviting target to shooters than his head and is the most heavily armored part of his body. In addition, some continuities included his cape as part of his armor, allowing him to extend a bulletproof and usually fire-resistant bit of protection to others.
The question should be why doesn’t Batman wear powered armor?
The answer is considering the power and strength levels of the members of the Justice League including, Superman, Wonder Woman, Captain Marvel, Martian Manhunter, Captain Atom to name just a few of the heavy hitters, there would be no point for him to wear a suit of armor that would make his strength a pale shadow of these mighty giants. His contribution to the Justice League is of a different caliber.
A Bat-Knight in shining armor
In light of the company he keeps in the Justice League, people have wondered why he does not wear a suit of more advanced powered armor to allow him to have an easier time in combat fighting his enemies, giving him superhuman strength and resistance to injury and extending his ability to carry more diverse weapons and tools?
Batman has been seen in a variety of iterations where he has worn a battle armor. Frank Miller, whose most famous armor-wearing Batman in Dark Knight, showed a Batman who was much older, whose physical capabilities had fallen and he was preparing himself to fight against the world’s mightiest hero. He knew he had no chance, but he needed to distract Superman long enough for him to use his matchless intellect to win the day.
Azrael was another iteration of Batman who used a suit of powered armor as part of a reign of terror on the underworld. While the suit was highly effective, Azrael did not understand the minds of the people he was fighting and ultimately failed in his tenure as Batman.
The most famous armor wearing Batman is the future Batman, Terry McGinness, genetic son of and trained by the aged Bruce Wayne. The battle armor Batman created was initially to continue his crime-fighting career as he aged but eventually even with the suit, his heart was unable to keep up with his combat-laden future. The enemies were more powerful, more numerous and he was simply unable to keep up the pace. Terry on a quest for vengeance, comes to the attention of Bruce Wayne and with the support of Wayne, Terry took on the role of Batman. Without the physical training of Wayne, Terry is dependant on the suit and the ghost of Batman whispering in his ear remotely.
Modern Batman doesn’t wear powered armor because he doesn’t need to. His primary contribution to the Justice League and to crimefighting itself has less to do with his weapons, tools or the addition of a suit of battle-armor and more to do with his peerless intellect, amazing powers of deduction, natural intuition and honed sense of the psychology of the criminal mind.
His matchless intellect is his superpower. His incredible drive, intensity and dedication are his calling cards. Anything else is just a tool for him.
Batman’s brilliance created protocols to destroy or immobilize every member of the Justice League and found those protocols to be deadly effective when they were stolen and used to great effectiveness by his opposite number, Prometheus. If this is what he does to his friends, what do you suppose he plans for his enemies?
It is his mind that allows him to be at the right place at the right time when a crazed, but highly intelligent Riddler or Joker plot some scheme their demented psyches demand they torment Batman with.
It is his mind that allows him to combat the ageless enemies, The Sensei, Ra’s Al Ghul or Vandal Savage who with their extended lifespans have amassed incredible skills and abilities that have almost no equal on Earth.
It is his genius, that gives him the ability to not only hold his own amongst the pantheon of godlike beings of the Justice League, but to direct those beings in ways of using their powers they themselves have never considered.
What Batman brings to crime fighting is his HUMANITY. His frailty, his understanding and memory of the human condition. The sense of being in the presence of giants and wanting to remind those giants that HE not THEY are the future of humanity.