Both sides of a 240 breaker should have 120V to ground and 240V between the two of them. How are you measuring this?

I just put my tester (the one that beeps when you put the plastic tip on a hot wire) on the front of the breaker, next to the switch.

It buzzed on the top one, but not on the bottom one. Buzzed on all the other breakers as well.

1. A dryer uses 240 for heating, both lines, and 120 for controls and motors, one of the lines.
2. When you replaced the breaker was there a tab or slot for both sides of the breaker? Try moving the breaker by swapping with another, can be as simple as pulling a single next to the double, move the double one space and place the single in the open slot.

Both poles of the breaker should be "hot". I'd suggest turning off the breaker and then turning it back on. It's not uncommon for one half of a double pole breaker trip without making it visually apparent that it is tripped.

If it trips again, then additional investigation is warranted.

A two pole breaker that doesn't take out both poles when either side trips is DEFECTIVE and needs to be be replaced.

His non contact tester is NOT a really great way of diagnosing things.

A non contact tester and even some digital volt meters are not all that reliable in crowded wiring.

A DVM's input impedance is often so high it can pick up 'stray voltage' by capacitive coupling, the same 'trick' that non-contact testers use.

Luckily the error is usually to indicate voltage is present when it is NOT actually there.

It can make trouble shooting a PITA though when it says dead circuits are still live.