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.
Well, first of all...you probably didn't have an actual 277volts. What you had is a cheap meter. I have seen all kinds of whacky voltage readings from the $5 harbor freight meters I have laying around.
The only place one sees something other than 120 or 240 volts in a "normal" residential setting is the very rare "3 phase power". The vast majority of homes do not have any 3 phase power available. It has to be run to the house specifically as an unusual setup. When this exists in a home it would add "208V" to the list of possible voltages. Again, very rare and probably not at your house.
The only place you would see a real 277V is in a big factory. It is normally used for HUGE machines such as big welders and bigger shop equipment than you have probably seen before. Normal DIY shop equipment runs on 120V. Big shop equipment runs on 240V. HUGE shop equipment requires 3 phase connection with the 208V "wye 3 phase" mentioned above. It is extremely rare in a house. The 277V delta 3 phase would never be seen in a house, and only rarely in any commercial building you have been in. It is typical only in large factories.
Now as for what's probably wrong...you probably have a loose connection or a broken wire on one side of your 220 connection. This would give you 0v between the two hots, but still 120 from 1 hot to neutral.
As for how to fix it? Hire an electrician. Could you fix it yourself? maybe. Could you burn your house down? maybe. Could you sleep at night after messing with it yourself? only just maybe. Hire a professional for anything that is a combination of dangerous and slightly outside your comfort zone.
Best Answer
Wild guess while we wait for details... You may have one of the rare panel brands that allows a dual breaker to be installed with both sides on the same leg of the panel. Try moving the breaker up or down one slot.