I am travelling from Chennai to Phoenix and I have a layover of over 9 hours in O'hare airport. I am travelling on a student visa. I reach at 19:40 local time and my flight to Phoenix is scheduled at 5:00 the next day. Is it possible to stay at the airport without checking into a hotel? Or will I be asked to vacate the premises?
Any question about staying overnight at an airport can usually be answered by consulting the comprehensive sleeping in airports website. It's probably best to read the reviews on that site to see the experiences of previous passengers there. Since you're arriving from an international flight, you will have to clear security, re-check your luggage if you have any, and you may or may not be allowed through security for your flight the next day. I suspect you won't be allowed through but can't confirm that.
If there has been disruption, apparently the airport sets up cots for passengers. If not, you will have to find a bench or quiet corner to sleep. If you can afford it, you'll obviously be a lot more comfortable in a hotel. Looking on google maps for hotels near the airport, there seem to be reasonably inexpensive hotels nearby in the order of $80-$100.