Will a microwave induce current in a transformer placed inside of it?
If a transformer, say your typical 12v wall wart, were to be placed inside a   microwave and then powered on, will the microwaves induce current in the transformer? 


This depends on many things, but any "wall-wart" variety may get hot, arc, and/or catch fire. So yes, the microwaves can induce curents in wires, but probably not how you think they will. And any wires from this leading out of the microwave, if the safety-interlock door could even be closed, could contain extremely dangerous voltages.


  What would happen if the transformer was also powered as well as the microwave?


The afore-mentioned. Transformers use magnetic energy... microwaves are RF (wave) energy. Magnetism interfering with RF is a technical possibility, however the tiny amount of magnetism produced by a wall-wart (at 50 or 60Hz) wouldn't do much to 1000W of RF energy at 2.45GHz.

A microwave oven typically employs about 1000W of non-ionizing RF energy at 2.45GHz to heat food. This RF energy (wave radiation) is generated by a magnetron "tube" inside, which is powered by a large microwave oven transformer (MOT) which converts the household mains AC into about 2000V DC, 0.5A to drive the magnetron. 2000V * 0.5A = 1000W. MOTs and magnetrons are very dangerous devices.

(Contrary to popular belief, 2.45GHz is not the resonant frequency of water or fat or anything else; water's resonance point is actually around 22GHz. 2.45GHz was chosen due to economy of magnetron production and limited free RF frequencies.) 

The more pointy any metal objects are inside a microwave oven, the more likely arcing will occur. A smooth metal object by itself like a spoon may do little, but a fork will induce arcing at the tines. Aluminum window screening and steel wool burst into flame. Since 1000W of energy is put into the system, sparks of thousands of volts are possible. So a transformer inside an oven may induce some current in the outermost windings, but the electrical plug is pointy so may arc, which could quickly turn it into a flaming mass.

If you wanted to couple into some of that RF energy, you would have to design and use an RF antenna, not a transformer. 2.45GHz frequency corresponds to 12.25cm wavelength, so any conductor longer than a tenth of this (1.225cm) will start to conduct RF energy. (This is why the holes in the oven door don't let out any microwaves - they are smaller than 1cm.)

So in summary, no, you cannot place a transformer into a microwave oven and get any useful power out of it. You might get a deadly shock and fire, however. 

If a properly engineered 2.45GHz antenna were used instead, then some of the power could be coupled to that and used to do something. But then you would have 2.45GHz AC at very high voltages, and few to nil electronic components are fast enough or robust enough to downconvert that into anything usable. At this frequency, only low-power detector Schottky diodes work well, but they can't rectify much power, and are quite inefficient.