Montresor describes the chains very briefly:
A moment more and I had fettered him to the granite. In its surface were two iron staples, distant from each other about two feet, horizontally. From one of these depended a short chain, from the other a padlock.
These chains must have been there for hundreds of years. The huge accumulation of bones in the catacombs show that the palazzo and the site is hundreds of years old. Montresor did not install these chains; he must have just found them while exploring the catacombs. There could have been only one use for such chairs. They must have been used by feudal lords for the same purpose for which Montresor intends to use them. Disobedient peasants had a way of disappearing and never being seen or heard from again. This sort of thing was common all over medieval Europe. A man could get thrown into an oubliette and, as the word suggests, forgotten. He would die of starvation and madness in the dark.
Throughout "The Cask of Amontillado" we see that there is an unequal relationship between the two characters. Fortunato is rich, Montresor is poor. Fortunato is egotistical, assertive, Montresor is humble, obsequious. The symbolism of chaining Fortunato to the rock wall and leaving him to die is that Montresor is asserting his superiority to his victim. Montresor is turning the tables on his adversary. He has him begging for mercy. He is doing just what feudal lords did to enemies in the past. Now it is Montresor who is the lord and Fortunato the vassal.
We might do better to ask what isn't symbolic about this punishment, because so much is symbolic.
It happens at festival time (when normal society is turned upside down).
He walls him away in the family crypt, like a bad memory or repressed desire…and he is that, since he is remembering him years later.
He stores him away to savor him, like a fine wine, the source of their disagreement.
There are references to him being a Mason (a member of a secret society); he shows that he is literally a mason, and part of a far more secret society, and so on.