I love iced coffee, and usually make it. If you don't know what it is, it is basically adding cola to a brewed coffee. When I brew coffee in my coffee press, there is usually enough space to add cola.
I'm not going to add the cola before it is brewed, but after it is brewed, in order to easily serve it.
Can this damage the container or the press part? (The glass and metal parts)
Interesting, I have never heard of this brewing method before!
What kind of press do you own...an Aeropress/French Press? Can you post a link to the model? I wouldn't think this would hurt the press. However, it is best that you clean it more over than you would with just normal coffee - Cola tends to have a lot of added sugar/high-fructose corn syrup that could dirty the glass/plastic and quite possibly attract bugs like ants.
Soda also contains different acids like Citric and Phosphoric. As we've seen on the popular show Mythbusters (and the internet alike) the acids can be used to polish chrome and remove rust. As long as you don't have a cheap metallic press, I wouldn't worry too much about soda affecting the integrity of the metal (especially since most metal presses are made from Stainless steel).
