Tirelire translates to "piggy bank". It's a charming and somewhat nostalgic word that isn't commonly found in song lyrics, making it stand out.
In the song, the narrator is overwhelmed by having too much small change. He sings, "Vous avez cassé votre tirelire" (You've broken your piggy bank), humorously suggesting that the sheer volume of coins he possesses is comparable to someone emptying a piggy bank, highlighting his exasperation with the endless stream of small money.
Charles Aznavour turns a small daily annoyance into a whole song: too much loose change.
The narrator starts out collecting coins from vending machines, listening to them jingle and clink all day until it wears on his nerves. He switches jobs to become a delivery man, hoping for a break, but tips arrive as more coins. His pockets fill up, wear through, and eventually tear.
He compares himself to people carrying the "beautiful notes" of the Banque de France, while he's stuck with pockets full of small change that ruins his clothes and his patience. It's a light, self-mocking complaint, using "so much change" as a running joke about a problem most people would love to have.