Raspi Audio Problems When Using a Servo
This is likely to be caused by shared noise through the power supply. There are three possible fixes:


Put the servo on its own supply (common ground, but different supply). Although it looks from that tutorial that it's already powered by batteries.
Use digital audio to another board with its own supply (I believe there are various Raspi DAC projects out there, or USB audio may work). You could even have optical isolation over TOSLINK.
Attempt to suppress the noise from the servo with chokes, decoupling capacitors, etc. Start with a few uF across the power rails near where the servo is connected.


You said you'd set audio to HDMI but were getting noise out of the audio jack. I believe if you do that you should get audio on the television or monitor used for the display, and you should disconnect the audio jack.