Yes for safety and to keep the Flaxseed oil from going rancid and bad the Flaxseed oil should remain refrigerated when not using it.
Even if you've never opened the Flaxseed oil it should be kept refrigerated to prevent it from going bad.
The Flaxseed oil goes bad pretty quickly even when not opened so refrigerated the Flaxseed oil lasts longer.
Once you open the Flaxseed oil it should always be refrigerated for safety otherwise it could make you sick if it's left out and goes bad.
If the Flaxseed oil has gone bad or has been left out of the refrigerator for longer than a day then throw it out to avoid getting sick.