In a USA 240 V a/c circuit there are two breakers (usually in a single unit called a 2-pole common trip breaker) connected in series on a single wiring path so that the current alternately flows out of one breaker, through the wire connected to that breaker, through the load (here the dryer), out of the load and into the other wire and finally into the other breaker. So one does not add the current flowing in the two breakers; there is only one current path flowing through both breakers. The two breakers must be on different legs of the bus for this to work.
If your dryer circuit is supplied by two separate 1-pole breakers which do not have a common trip and if the two breakers are on separate legs of the bus (i.e., 240 V between the two breakers), then the dryer would work, but this would not be according to code and could lead to a dangerous condition or could possibly damage the dryer (not sure about the latter).
One could connect two breakers in an arrangement in which one would add the current in the two breakers. However this is not used because it is pointless, dangerous, and not according to code, but I will describe it so that you can better understand why you do not add the currents in the case you are asking about.
You could connect separate wires to two different breakers which are on the same leg of the bus (i.e., zero volts between them) and then connect those wires together with a third wire to go to a load. From the load another wire would go back to the panel but to the neutral, not to another breaker.
In this arrangement, the current passing through the third wire and supplied to the load (and carried by the neutral wire back to the panel) would be the sum of the currents through the two breakers. So if the current in each breaker was say 25 A, then 50 A would be supplied to the load. (Also the voltage of such a circuit would be 120 V, not 240 V as in the standard case.) But as I said you will not encounter this arrangement because it is not something that accomplishes anything that cannot be done safely and according to code.