Once the engine warms up it shuts down. there is some backfiring every once in a while to. We have rebuilt the carb, replaced the pet **** and replaced the tank. Any suggestions on what to check next?

Check for vacuum leaks. Spay starter fluid around intake,carb base, etc., while engine is idling, listen for change in engine idle.
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clean out carbs set float levels to correct height.

Most engines hav an anti backfire valve solenoid screwed under the carb bowl. This sometimes sticks due to ethanol gas. When you turn off the ignition, it closes the valve to let the engin run out of fuel. If it sticks closed the engine will not start. Sticken open, the engine will not shut down. It has a single wire going to it. When you turn on the key, you should get power to the solenoid. If you have power to it, then remove the solenoid and clean the needle valve and carb opening. Would be a good time to clean carb also. A good carb cleaner works well. Ethenaol added gas is a real culprit to carbs nowadays.

You might try changing the fuel line itself. Do this especially if anyone has recently changed them. Though heard of less these days than in the past, some fuel lines will separate and actually close up somewhat once it gets warm. This is usually accompanied with a significant bit of backfiring and popping.

Check the slow running jet for glogging, set the mixture screw 2 turns from fully closed and make sure the small filter connected to the fuel tank is not blocked.

Re-set the idle adjustment screw on carb? Also check fuel / air filter.

Have you check to see if the timing on the car is right?

Is the float sinking & causing a rich condition?

Was it replaced during the rebuild or not set per
the rebuild kit instructions?

It seems that you have a vaccuum leak around the intake or carb, Check all hoses for good connections. You can spray carb. and choke cleaner around the intake and carb whole it is running and if the engine rise and lowers. you have a leak

It would appear that the Black smoke is unburnt fuel, meaning Spark or timing could also be out. Check you plug and point gaps. If you rebuilt the carby, float level is crucial also check the needle and set to ensure it is shutting off the fuel, if it is not shutting off, it will just flood the engine. Not sure what carby you have but I woud pull out the jets/power valve and replace them. Some carbys have a small ball bearing to control fuel, if that was lost that could be it. Pull up an exploded diagram of the carby and do it again. I know its time consuming, but it will be peace of mind. Good luck