PC Power Supplies are switching type circuits and use the various coils in their design to support the conversion of the line voltage to the output DC voltages. NO you cannot just go in there and start yanking out random coils.
Are you sure that this power supply is capable of supplying the current required to drive the cordless drill. Cordless tools can draw rather high current levels from their batteries and many cheap computer supplies are just not going to be able to support supplying the current required.
Another issue.... the speed control on cordless tools usually operates by switching the load current to the tool's motor on and off at a high rate. The normal tool battery can be quite fine with this behavior. On the other hand the cheap PC power supply will likely not like the current surges from switched motor load. Especially if you have already heavily loaded the PC supply this switching could be what is causing the noise to come out of the unit. The load switching frequency and the power supply switching rate could even be interacting with each other making the problem worse.
My suggestion is that you properly determine if the power supply is rated to provide the current required for the tool. Then if so add some large value capacitors to the 12V output to GND to help counteract the switching surges of the motor load in the tool. Take a look at something like these caps to get an idea of what I am suggesting.