If the power supply doesn't give enough current, the power supply might overheat or simply not give enough to turn the device on.

The NES power supply gives out 9V AC, which would be bad for something which runs on DC. Does the Famicom run on DC?

These are the things to verify for a power supply:
- wall voltage generally matches power supply input voltage
- output voltage matches device voltage
- power supply outputs enough current for device (more is fine)
- DC power supply of proper polarity for DC device; AC power supply for AC device

DC is often indicated by a tiny picture like this:

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- - - -

An AC device can sometimes be run on DC if it has a rectifier as the first thing, like the US NES. I spliced in a second power connector to my SNES power supply which I use to run my NES and SEGA Genesis (eliminates extra wall transformers for each). I think the SNES power supply outputs 10V, but the NES and Genesis have voltage regulators and sufficient heat sinks that the extra volt doesn't matter.