Can i burn my catalytic converter by running the car with broken exhaust?

95 Ford Escort has a cat converter connected directly to the exhaust manifold in the engine compartment with the elbow coming down and connecting to the exhaust pipe by a braided flex, to a center resonator to muffler and tail. The O2 sensor is also connected right at the manifold cat converter connection. Where the braided pipe is, the pipe has completely broken by rust.
I have driven less than 50 city miles since. I just got new parts, but need to drive 45 highway miles about 45 minutes to get to where I am going to work on car.
On a recent “Car Talk” show, they discussed driving a car without exhaust pipes and said the caller had burnt the valves and rings on her car because of lack of back pressure in the exhaust.
I think the Cat Converter is more sensitive to overheating, AND DON’T KNOW how OLD it is! The CAt Con is an additional $ 170 ! Not to mention the extra work of trying to remove those heat rust frozen nuts from the Manifold studs. The local Pep Boys wants $ 100 labor to install my parts. I am out of work and would like to save and do it myself. The heat shield has already rusted off the Cat Con. The Haynes manual says once the heat shield falls off due to rust, replace the CAt CON; but the parts guy said if I can pass inspection as long as the inside is working don’t mess with it.
Any advice???!!

No risk of burning your catalytic converter by running like this. Very slight risk of burning your valves by running like this. Your catalytic converter provides a decent amount of back pressure, probably enough to prevent damage to your valves. Some people drive like this for months or even years without damaging anything. You can probably get away with this, but do know there is some risk of damaging your engine. Burnt valves are usually the result of running your exhaust wide open, as in, open headers or less.

Keep your windows open, you’re more likely to poison yourself with carbon monoxide than anything else.

This should be replaced solely due to the danger of carbon monoxide poisoning if for no other reason.

As to the cat converter, you cannot burn it up, but you can cause it to experience an early demise. The converter itself is a metallic (platnum-palladium) coated ceramic honeycomb capable of withstanding high heat. But if the leak is allowing exhaust to escape prior to the upstream oxygen sensor it could cause the engine to run rich, and that would cause carbon to deposit on the catalyst itself. The carbon could also deposit on the surface of the oxygen sensor and cause it to also lose function. If you have an OBDII system (I don’t know about the '95 Escourt…OBDII systems weren’r required until '96), you’ll get a Check Engine Light.

As regards the cat converter heat shield, loss of the shield doesn’t affect the operation of the cat converter but it could be a fire hazard.