If the furnace starts intermittently, though it runs fine when it does start, the problem is likely the furnace control relay running from the thermostat to the furnace. Unfortunately the Lennox Elite Series furnace is known for this problem. It's a simple relay box attached to the side of the furnace with two wires running to the thermostat. There should be a few lights on the box as well. If the lights do not come on, then the furnace relay circuit board is broken. If they flash on and off, then there's a loose connection in the wiring, either running to the thermostat, or passing into the furnace itself.
Furnace doesn't always start when it's supposed to
Furnace is too noisy
First check the access panel of the furnace. If it's loose, it can rattle and create a rumbling noise throughout the entire house. Next check the fan belt inside the access panel. Some lubricant spray might be in order, though if the fan belt is frayed it should be replaced. Next check the blower motor; it may be in need of oil, which should be fed through the induction ports on the base of the furnace unit. Finally the noise could be caused by dirty burners. If you're using old oil, this can cause the burners to become dirty faster, though they should usually be cleaned by a technician once every five years.
Blower won't stop running
If the blower motor won't stop running regardless of the actual temperature of the home, there are two possible causes. The blower control could be incorrectly set, or the limit switch on the Lennox Elite Series furnace unit itself isn't set. Go to the thermostat and switch it to the AUTO setting, then go to the furnace unit itself and reset the limit switch. You may need to indicate the time and temperature threshold which you will allow the blower to run before it automatically shuts itself off. If you've reset the limit switch, but the blower still won't stop, then the limit switch is broken and needs to be repaired.