circuit breakerelectricalelectrical-panel
After looking at it more carefully:
Updated with more pictures.
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.
Which ties the breaker handles together, to provide common trip characteristics.
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.
Use a multimeter to measure voltage between the two hots (black to red). You should get somewhere around 240 volts. If you get 0 volts, the breakers are on the same leg.
An empty service panel looks like this
I've drawn rectangles where breakers would go, and labeled the legs A and B. As you can see, breakers across from each other (left/right) are on the same leg, while breakers next to each other (up/down) are on different legs.
If you measured between breaker 1 and 2, you'd get 0 Volts. This is because they're on the same leg, so they're at the same potential. If you measured between 1 and 3, you'd get ~240 Volts. These breakers are on separate legs, so they're not at the same potential.
Best Answer
Easy. For each wire, determine if it's 10 AWG, 12 AWG, or 14 AWG wire. Then consider the following table:
Don't upsize any breakers. But if the breaker is bigger (in amps) than the wire in the above table, downsize the breaker. BR breakers cost $5 for a single and $10 for a double.
Any wrong-type breakers (looking at you Square D), replace them.
If any cables have 2 hots (e.g. red-black-white Romex) get a 2-pole breaker of appropriate amperage and land both hots on the 2-pole. Note a 2-pole is not a duplex. A 2-pole is twice the size of a normal breaker.
That's it. You're done.
After this, some loads may trip the breaker. If that's the case you'll have to follow the usual troubleshooting - too many loads on one circuit, too small cable needed, etc.
If a breaker serves only one motor load, ask a new question with the motor specs and ask whether you can use (size of wire you have) with (size of breaker you used to have).