It sounds like you may be mistaken as to how this is wired, or that perhaps I'm just not understanding your explanation. As others have mentioned, it's not possible to get 240 volts from a single pole in a 120/240V split phase system. Each tandem breaker provides 2 120 V circuits, this is true. However, if you measure between the terminals on a single tandem breaker, you'll get 0 volts. This is because the terminals are both powered from the same leg, and so are at the same voltage potential. If you measure from a terminal on the top tandem breaker to a terminal on the bottom one, then you'll measure 240 volts. This is because each breaker is connected to a different leg, which are each one half of a 240 volt circuit.
With all that said. For this setup to work, one appliance would have to be connected to both breaker. Something like this...
Notice that each appliance circuit has one wire connected to each of the tandem breakers. In this situation, you'd need a device like Speedy Petey shows.
![common trip device](https://i.stack.imgur.com/p304Y.jpg)
Which ties the breaker handles together, to provide common trip characteristics.
![enter image description here](https://i.stack.imgur.com/A9CJW.png)
Notice how the inner handles are tied together, and that the outer handles are also tied to each other. This way if either trip (or are turned off by the user), the entire circuit is shut off.
If this is wired the way you've explained, where the dryer is connected to the top tandem and the heater is connected to the bottom. Then there's some magic going on in those breakers.
Was the motor replaced with the same size (horsepower, amperage, voltage) motor? What is the amperage and voltage listed on the motor's data plate? In many cases the motor plate will either list the size of the required breaker, or if not, the manufacturers website will have that information available. If you "had the motor changed" it might be best to call the company that changed it and report that the new motor is tripping the breaker, where the old one did not, and have them check for issues, since they should have verified that the new motor would work the same, or made any changes needed to support any change in the motor. If you changed it yourself, that's on you.
You should not replace the breaker with a larger breaker unless the new motor requires a larger breaker.
You should not replace the breaker with a larger breaker without ALSO replacing the wire with larger wire, suited to the size of the larger breaker, which should be suited to the size of the new motor. 20 amp, 12 gauge copper, minimum - 30 amp, 10 gauge copper, minimum.
Rather than "sanding the wires" you should (with the breaker off, of course) cut off the damaged ends of the wire, pull it a little further in, and strip the ends to get new undamaged wire. If this is not possible, consider replacing the wire, especially if it's fairly short run.
110V (single-pole breaker?) is rather abnormal for most well pump installations. One potential fix would be to re-wire for 220V operation if possible
Best Answer
Update the panel, it needs it.... An electrician needs to do this.
The breaker simply cannot be changed out to a higher rated breaker, the wiring has to support it too. This is what a qualified electrician can do.
The electrician can tell you if it is possible for it to be done with your existing wiring.