If you cut onions and leave them out on the counter or at room temperature then the onions can attract bacteria which then makes them poisonous which could make you sick.
If you cut onions and plan to use them the next day then make sure to place them in a sealed container or ziploc bag and then refrigerate them to prevent bacteria from forming on the onions and the cut onions will be safe to eat the next day and up to 7 days after being cut.