First a comment: the use of the power&ground rails on the right side for the beeper is confusing at first glance. It's best (and clearer) to keep those for power & ground.
Next, you didn't solder the header to the TPL board. Anything in a through hole pad needs to be soldered to ensure a stable, quality connection. If it's not soldered it won't work, very well or for long. It's almost guaranteed that a high speed data signal like in an I2C or SPI connection (not the case here) will just not work.
The real problem is that the TPL5111 doesn't work how you want/need it to.
The potentiometer sets the timing, i.e. have it beep every 30 seconds
The duration of the beep is controlled by the DONE signal. DONE has to be usually low, i.e. 0v. When DONE is switched to high (i.e. the voltage on the VDD pin) the output "turns off" and will remain so until the time set by the potentiometer has elapsed, and the whole thing repeats. Your problem is that you aren't using the DONE pin to turn it off.
You can use a simple RC (resistor/capacitor) circuit to make DONE high (enough) after a short time. That time will depend on the values of the resistor and capacitor.
Wire the resistor between ENout and DONE, and the capacitor (paying attention with which side is negative (connect it to ground)) between DONE and ground. Now with ENout goes high the capacitor will charge through the resistor and the voltage at DONE will increase. When it gets to a high enough value, DONE will be triggered and ENout will go low again (discharging the capacitor as well as turning off the buzzer). I found that a 27k ohm resistor and a 10uF capacitor gives about a 1 second beep. Using a higher value resistor will cause a longer beep.
See issue 10 of Hackspace Magazine (free PDF at
https://hackspace.raspberrypi.org/issues/10) for my article on capacitors.
Dave