Metro tickets are requested at any metro station in the Capital Wasteland that still has active protectrons. Having a ticket can prevent making one of these robots hostile. These protectrons are inactive by default, but can be activated by hacking a nearby terminal as they then come out of their capsule and search the area for hostiles and passengers. When a Protectron is activated, it will approach a target and scan for a ticket. If it finds one on them, it will take the ticket and let the person move along. Therefore, it is advised to collect as many tickets as possible if the plan is to activate the Protectrons to help clear the metro. However, as the Protectrons are slow walkers, it is possible to simply avoid one if not having a ticket by falling back before it gets a chance to scan for one, avoiding combat.
Metro tickets aren't found in abundance around the wasteland except in Metro stations, but with the add-on Broken Steel, as many as 50 tickets can be found scattered throughout the Olney Powerworks. In one section of the Powerworks, there is a desk with a stack of Metro tickets that are assembled in a traditional triangular card stack.
Some feral ghouls in the Metro tunnels carry tickets, implying that they were passengers before radiation transformed them into ghouls.
Xbox 360 When the protectron scans the Lone Wanderer for a ticket, it will not detect the ticket(s) in the inventory and will go hostile. [verified]