Is the shirt considered impure if my friend touched it after he touched a spot in the pharmacy that was touched by a young man said to be homosexual, and must the shirt be disposed of, or is wearing it considered among the acts of the people of Lot?

This young man did not become impure by his disobedience, for his body and whatever is connected to it are pure unless an impurity is present. Therefore, there is nothing upon you regarding what you mentioned, because the default state of things is purity, and as long as your friend's touch did not involve the transfer of a physical impurity from his hand to your shirt. It seems that you are experiencing obsessive doubts (waswas) regarding matters of purity. The solution for this is to refer back to the rulings of Sharia concerning them, while frequently seeking forgiveness from Allah, remembering Him (dhikr), and relying upon Him.