Should I install the jumper on my new limit control?
The limit switch is a normally closed switch that opens on a temperature rise.
The fan switch is a normally open switch that closes on a temperature rise.

Think of them as 2 separate switches that are side by side in a common enclosure.

Each switch needs a power supply and each switch lets that power run through it under the proper conditions.

They both sense the same temperature inside the furnace.
So when the burner kicks on and the temperature inside the furnace rises it causes the fan to come on. But if the temperature inside the furnace gets to high (blower motor fails, blower belt breaks, filter dirty so air flow is low etc) then the Limit switch will open and cause the burner to shut off.

You need the jumper that will put 120 volts on both INPUT sides of the switches Fan/Limit control.

That way as the furnace heats up, the normally OPEN fan switch will close it's contacts to turn on the fan motor and the normally CLOSED limit contacts will remain closed to let the power through the safety circuit and only open in the event the furnace gets to hot and will cause the gas valve to close.

Now it is important to note that from your description of your issue that you have a high voltage limit control. Some furnaces use a low voltage limit control, in that case you do not use the jumper.

I just noticed that your question is 2 years old....duh me..... well I hope this helps anyone who has the same issue.