You do need to have elite status with the airline to receive a complimentary upgrade.
However, you may be able to purchase an upgrade when you check in for your flight or at your departure gate.
You may also be able to use your frequent flyer miles to secure an upgrade. You do not need to have elite status to purchase an upgrade or use miles for an upgrade.
If your flight is disrupted and you are required to take a different flight, you can ask for an upgraded seat to compensate for your inconvenience, and your request may be granted if space is available.
Another opportunity for an upgrade request to be granted for a non-elite is when you volunteer your seat because a flight is oversold.