So Pat, the OP has not been seen since the day they posted, in Oct 2013. Hopefully this will be of help to new viewers
I had to comment that I highly doubt they have 14 GFCI breakers in their panel. I would all but guarantee that they are referring to AFCI breakers, and they are experiencing nuisance trips. Could even be that the breakers are the early SqD AFCI breakers that were recalled. This was very common of early AFCI breakers which is why you see so many subsequent incarnations.
I'll also add that grounding and bonding, or the absence thereof, would have NOTHING to do with nuisance tripping of AFCI or GFCI breakers.
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.
Best Answer
This has happened to me. You may have just manually tripped the breakers when you were putting the cover back on. It doesn't take much force to trip the breaker.
It's unlikely that an electrical mis-wiring caused this, because each side of the box is fed alternatively by both of "hot" feed. Say you have split phase 1 and 2. The left side of your box will be fed by phase 1, 2, 1, 2, etc. The breakers on the left side have nothing in common with each other that they don't also have in common with the right side.