What would happen if an airplane flew through a tornado?
It depends on the strength of the tornado and other variables, but in most cases, a plane Vs tornado will end very very badly for the aircraft and its occupants in a low altitude high speed scenario.
A good example would be the NLM CityHopper Flight 431, a Fokker F-28 that flew into a thunderstorm 8 minutes into takeoff and crashed after the wing was torn off. From the summary of accident report at aviationsafety.net:

At 17:12 the aircraft entered a tornado, which resulted in loads on the airframe increasing to +6.8 G and -3,2 G. The right wing was bent upwards followed by a severe downward sweep. This compromised the structural integrity of the wing, causing a large portion of the outer wing to separate in an upward and rearward motion. Control was lost and the aircraft impacted a railway bridge inverted.

Everyone abroad the aircraft perished. FAA is pretty blunt about flying near one:

Pilots should observe the following rules for any flight routed even potentially near actual or possible thunderstorm activity:
• Avoid all thunderstorms.

And also,

Although pilots have flown through limited state thunderstorms with little or no damage to the aircraft or passengers, these pilots should consider themselves extremely fortunate to be alive.

(emphasis original).
Note: I've used both thunderstorms and tornadoes here. weather.about.com defines them as:

Thunderstorm: A storm (produced by a cumulonimbus cloud) that includes rain showers, lightning, and thunder.
Tornado: A violently rotating column of air that extends down from the base of a thunderstorm to the ground.