Electrical – Why is the dimmer switch buzzing

The dimmer switch has some solid-state electronics in the path between the electrical mains and the light that it controls: definitely some form of transistor, and most likely one or more MOSFETs. MOSFETs have very high resistance when they're off, well into the gigaohm range (or higher), and very low resistance when on, in the range of 1 ohm (or lower). Other transistors types have a less extreme swing between on- and off-resistance.

The rest of the circuitry in the dimmer is designed to operate on AC mains voltages, so the small DC voltage that your multimeter drives through the dimmer isn't enough to turn on any of the transistors in the conduction path, so you're reading it as being open-circuit.

If wires spark enough to generate a burning smell, you'll hear the arc and it's also possible the breaker would go as well.

From the symptoms you describe it sounds like the dimmer itself is going. Since it has a burning smell, I would say it's unsafe to use and you should replace it immediately.

Of course, if you see any nicked wires, or any of the wire nuts are loose then you should fix those problems. Be sure to turn off the power at the panel when you're doing this (you should check with a non-contact voltage detector just to be safe).