Why would my clothes drying machine's switch keep on getting fried?
In the first place, having two pieces of equipment (the dryer and the aircon) that both need quite a bit of power, both on the same circuit is probably not a good idea. You have already noticed there is interference between them. So the first step would be to get one or the other onto a second circuit. I don't know how electrical codes are in your country (which you haven't specified - it may be significant due to different voltage levels), but in most of Europe we tend to put:


lighting on one phase
the kitchen plaque (a high power consumer) on a second phase
power outlets on a third phase.


The name of the game is to spread consumption equally across all three AC phases. 

In the second place, you say the dryer is rated at 25A. It would be nice to verify that it is really drawing that much current but no more, for example with a clamp ammeter (e.g.: http://en-us.fluke.com/products/clamp-meters/ ). Perhaps somebody in the trade could lend you one.

If it really draws that much current (or almost as much), this is indeed a piece of equipment with a high power requirement, that I feel should go on its own circuit all by itself.

Additional note: there are two kinds of electrical "tripper". One is the classical circuit breaker, that breaks the circuit when excessive current draw takes place. This is what happens when you have a short-circuit within the apparatus. 

The second type is a differential breaker, that actuates when a large difference in current takes places between the phase and neutral wires. The idea is that the difference in current must be going somewhere - possibly directly to the machine chassis and then to earth. This is quite dangerous for the human operator, since if he/she has wet hands and a good connexion to earth, the excess current could very well short to earth through the operator. 

It is standard practice (at least in Europe) for several circuit breakers to be protected by a single differential breaker. If any of the circuits develop a short to earth, the differential will trip leaving all circuits open.

The OP would need to have the dryer checked, that it has no internal short circuits. This can be done by connecting the apparatus to another circuit.

He should further check it has been correctly wired to AC. Swapping the neutral wire with the earth is a relatively easy mistake to make if non-standard color wires have been used. Hint: the earth wire and connexion should be green/yellow and have a slightly larger section than neutral and live. 

HTH