"I exit the bank."
The particle に implies movement and direction. If you said "銀行に出ます", this imples you're leaving to go to the bank. を lacks the implication of direction. You'll see similar use of に and を for boarding/exiting trains and buses.
Ok, but in the case of entering, both should be allowed?
銀行に入ります (I enter towards the bank, bank is destination)
銀行を入ります (I enter the bank, banks is object entered)
This should only be a problem for 出ます?