Checking continuity using multimeter
The continuity test function on a multimeter is essentially a resistance measurement - the continuity buzzer will sound if the measured resistance is less than some fairly low value - under 40 Ohms for one of my meters.

When measuring resistance, a multimeter will apply some voltage between its leads, and measure the resulting current to determine the resistance (or apply current and measure voltage).  If something else is also causing current to flow between the test leads, that current will affect the measured resistance, as the meter can't distinguish between the current it is providing and other current in the circuit.

Therefore, you must never do a resistance or continuity test on a powered circuit.

Also it is important to note when measuring components in-circuit, that the meter will report the effective resistance between the test leads, which may include other components in parallel with the component you are intending to measure (and touching both probes with your fingers can also upset the reading, due to the resistance of your body)