What happens when a TVS diode "burns"?
TVS diodes (Transzorbs) are semiconductors, essentially 2 zener diodes connected back to back. However, a zener diode is designed for continuous current, while Transzorbs are designed for transient surges of high intensity (5kA to 15kA typical) but short duration (&lt;20uS). Typical test current for clamp voltage is 1.00mA constant current, regardless of clamp voltage or body size-at least where I worked at. Pre-fail mode is evident by an increase in clamp voltage. A drift (permanent) more than +5% means it should be replaced to avoid latent failures.
The accumulation of heat in the form of ambient plus constant current plus transients combine to melt and/or burn the wafers into one short circuit. Their fail mode is to short or explode off the PCB if the follow current is high and is not shut off by fuses or limited by series resistors.
Because they have a much lower capacitance than MOV's, and have a much sharper avalanche, they can be and are used to protect digital logic and analog inputs very well, especially if a series resistor with the input of 100 ohms or more is possible.
There are some tiny smd versions with a capacitance of only 1.5pF that can be use in RF couplers and surge suppressors to about 2GHZ. They have the same fail mode as their larger cousins, but RF coax is often well grounded and protected.
The same cannot be said for test or ATE equipment. TVS devices are crucial to block connection and signal transients to inputs with sensitive semiconductors, and as back-EMF suppression for solenoids, relays and motors.
Because they fail so fast, fuses are seldom used to protect Transzorbs. Resistor values offer the best protection, if they are kept below a value that negatively affects signal integrity or needed bandwidth. MOV's are better at absorbing much higher surges for longer durations, and are trusted to protect equipment at the AC mains, even if it is 3-phase 4,160 VAC for well-pump motors. Containing them in fine quartz sand helps dissipate heat and traps debris from exploding devices.


EDIT: This answer pertains to TVS devices (Transzorbs) only. Other surge suppression devices such as MOV's, gas discharge tubes and sidacs have different pre-fail modes, though MOV's and sidacs finally fail by shorting. Gas discharge tubes fail by cracking open under constant fault current state, so the final state may be to fail open, or burn and cause arcing from soot.