From the Super Mario Wiki
The red medallion is a key item in Mario & Luigi: Paper Jam.
While Mario, Luigi, and Paper Mario are first exploring the Twinsy Tropics Dungeon, they come across a large gate with a Bowser emblem in the middle. Luigi thinks they should remember the spot because it might be important later on, while Mario thinks the gate is somehow connected to Bowser's Castle. They decide to revisit the gate later on, and continue through the dungeon.
After the trio defeats Papercraft Bowser Jr. with Papercraft Peach at the top of Mount Brrr, Bowser Jr. and Paper Bowser Jr. run away, dropping the medallion, referred to as "a weird object that the Bowser Jrs. dropped". As they chase after the Bowser Jrs., Kamek and Paper Kamek fly them back to the castle, causing the trio to fall off Mount Brrr. After Paper Mario saves them, Mario remembers the medallion and takes it out. Starlow then quizzes each of them about where they saw the symbol before, with the correct answers being in Twinsy Tropics, in the dungeon, on a suspicious wall. They remember the Bowser gate and decide to head back to Twinsy Tropics Dungeon.
As soon as they enter the dungeon, the Paper Goomba following them asks to see the medallion, inspecting it closely to make sure it is not counterfeit. He then betrays the group, shouting to the Fire Bros that they have brought back to medallion. He then reveals that it was all part of his plan to get Bowser to forgive his past mistakes and get a promotion. They must then get back the medallion by finding the Fire Bro holding onto it.
After they get back the medallion, they head to the gate and place the medallion onto it. The gate lowers into the ground, revealing a wall which takes them higher into the dungeon, rather than a direct path to Bowser's Castle.