Poe deliberately shows in many ways that Montresor is French and not Italian, although his family may have lived in Italy for a few generations. The name Montresor is conspicuouly French and means "My Treasure." As the narrator, Montresor speaks disparagingly of Italians in the third paragraph. He says, for example, "Few Italians have the virtuoso spirit." He does not participate in the carnival but wears a black cloak called a "roquelaire," which is a French cloak named after a French nobleman. He uses French words such as "rapier," and "flambeaux." He gives Fortunato two French wines, Medoc and De Grave, when they are underground.
All the human bones cannot have belonged to members of Montresor's family. They may date back as far as ancient Roman times. This is a puzzle. He or his immediate forebears may have acquired the palazzo only recently and have never removed the bones because there was no other place to put them. There may have even been a law against moving them. In fact, Montresor may not even own the palazzo but rents it. These old Venetian mansions could be rented cheaply because few people wanted the upkeep. They are relics of a distant time when the city enjoyed greater wealth and prestige. Montresor may feel he needs an impressive home because he apparently does business with foreign "millionaires," as does Fortunato.
The fact that Montresor is French may be a handicap socially and in business. He may be considered an outsider, a Johnny-come-lately. He is obviously struggling to survive. Even his servants do not respect him--possibly because he hasn't paid them. Fortunato, by contrast, is "rich, respected, admired, beloved; you are happy as once I was." His French background also tells us that the human remains cannot be of the Montresor family.
Why, humans, my dear friend! Mwah-ha-ha-ha-cough, cough. Sorry about that. Poe always brings out the beast in me, especially this wonderful story.
More seriously, these are the remains of the Montresor family—the narrators ancestors. While you or might find this creepy (well, at least you, mwah-ha-ha), it was not uncommon in some periods and locations to store the dead in catacombs, rather than burying them in individual graves. While some of the bodies might be there due to (say it softly) foul play, it's also possible for them to simply have died. What they are evidence of, though, are the way that Montressor hangs onto the past.
PS: For pictures of a catacomb, see the link below.