I haven't seen the diagrams of the 720 but here's what I'm thinking:
The voltage sense is for the battery charger. It sense the voltage at the battery itself, not at the DC input. The goal is to charge the battery as fast and accurately as possible. In order to do so, they force a higher current in the battery when it's power level is low. When the battery recharge, it's voltage will start going up. The sense wire picks-up that voltage increase and will signal the power supply to turn-down the juice.
A Zener diode work like a regulator and is designed to work in over-voltage conditions. A Zener is also designed to be installed the opposite way of a normal diode, in serie with a resistor. (
Diodes) If you installed it the "normal way" with the cathode (the white bar) towards the (-), it's no go. It must be connected with the cathode (the white bar) towards the (+) voltage.
A 5.8V Zener will simply keep the voltage at it's pins to 5.8V, no matter the input (to a certain limit, of course). The Zener is an absolute limiter and should be ideally installed in the power adapter, where size doesn't limit the maximum dissipating power as much as when it's surface-mount. Installing it in the device is poor design, IMHO.
If the Zener blew-up, chances are the power adaptor got faulty. I don't think it should ever output the 13.8V ever, even in open loop mode with the sensor wire disconnected. I would expect maybe 6V, not more.
I have read somewhere that some people simply open the power adapter, cut the white wire and solder the pad where the white wire should connect with the pad where the red wire connect, at PCB level. DO NOT FORGET TO CUT THE WHITE WIRE!! I have not done this personally but, what it does is limit the output to what's considered by the adapter as a fully charged battery so, the charging will be slower and possibly not 100%. Since the TT720 already have a very poor live time on batteries, that won't change much in real life and won't damage anything but it will prevent problems with the white wire breaking at the plug level: no more overload!