There is no luggage storage at the Colosseum in Rome
You have to pass through the security and if you have very large backpacks or oversized handbags, you will not be allowed inside the Colosseum. You should leave your luggage in your hotel room.
Also, the Colosseum management is strict about their no backpack policy. The website reads "We inform visitors that inside the monument it is strictly forbidden to enter with backpacks, handbags and luggage because of the new safety measure; people with reservations are kindly requested to show up at the ticket office at least 30 minutes before the time previously agreed."
Also, they do not allow water bottles. However, you can take an empty bottle and fill it up at the water fountains found inside the Colosseum.
Security lines and luggage rules apply to all city pass holders
Every visitor to the Colosseum needs to go through strict security. You can skip the ticket lines but not the security check. This applies to all pass holders: whether you have a Roma Pass, Omnia Pass, Rome City Pass, or any other pass for Rome.