The humor in Tellson’s description comes from the fact that it is a terrible place, and seems to take pride in being awful. Tellson’s Bank is described as “the triumphant perfection of inconvenience” (2:1). It is “an old-fashioned place” that is small and dark. The people there seem to prefer it this way.
They were even boastful of its eminence in those particulars, and were fired by an express conviction that, if it were less objectionable, it would be less respectable. (2:1)
Everything is small, damp, crowded and musty. Even the money seems to be moldy. Even the men are old (if there are ever any young men, apparently they hide them until they are old). Tellson’s is out of touch, and old-fashioned.
Part of the humor comes from the disconnect between Tellson's and the people, and part from the bank being an exaggeration of what most of us do not like about banks. We see the stuffiness taken to an extreme.