If your burner light is staying on, it could be for a few different reasons. The most common reason is that the pilot light is out. If this is the case, you'll need to relight the pilot light. Sometimes, the problem is as simple as that.
Another possibility is that there is a problem with the gas line. If the gas line is blocked or damaged, gas can't flow to the burner. This can be a serious problem that requires a professional to fix.
Finally, it's also possible that the thermostat is set too low. If the thermostat is set too low, the burner will stay on all the time in an attempt to reach the desired temperature. This can be a waste of gas and money, so it's best to adjust the thermostat if this is the case.