I will be traveling and arriving ORD later in the day around dinner. Because I will be going straight to a hotel by taxi that has no restaurant, I want to eat at ORD. If I do not go down immediately to claim my bad and eat upstairs what will happen to my checked bag? Assuming no other passenger takes it, will the airline just take it and put it in the office, or will this be a TSA issue? If this varies by airline, than I would like to know for American. Can I tell them when I land to hold my bag in the baggage office since I will be going to eat?
If you don't pick up your bag, it will be left on or by the belt for a little while, then taken to the baggage desk as an unclaimed bag. Eventually they will send it off to their storage area, but as long as you get there to claim it within an hour or so of landing it likely will still be at the desk.
Unclaimed bags typically are collected and go to a special handling area where they're stored for a while before being seized and eventually either destroyed or auctioned off.
That's not specific to ORD, it's common practice all over.
But why do things like that in the first place? Why not get your luggage and then go get something to eat? There are no doubt many places at the airport to get a bite. No need to risk losing your luggage and/or getting stuck on the wrong side of a security barrier from it.