Damaging a low voltage, 3-phase heating element possibe due to unstable voltage?
Based on change of element resistance with change of temperature your system may be OK unless you have other information not given here that indicates elemement damage. 

Why do you think the element is damaged?
Arcing will often cause damage, but the change in resistance reported may be due solely to the thermal coefficient of resistance of Tungsten - see below.



You say that the furnace reached 2000 C at 4V x 480A and that Tmax is 24500 C.
You mention applying 8V - what is the maximum rated element voltage? 

4V x 480A = 1920 Watts.
Your reported 8V x 390A = 3120 Watts. 

IF  temperature rise above ambient is dependant on energy loss through insulation then delta T ~ proportional to Wattage. This may not be the case, but what power/temperature law do you expect, and why?
(Note that the rough calculations below based on element resistance do NOT support this linear delta-T with power model). 

As Wattage is 3120/1920 = 1.625 times higher then if the essentially linear power-temperature relationship suggested above applies, then you might expect temperature to be somewhere over 3000 degrees C. 

The suggested upper heating temperature for Tungsten mesh in inert atmospheres is 3000 C and commercial furnaces are often rated to below that - eg see the Oxygon brochure referenced below. 

Even if a furnace has been designed to operate at the temperatures you are achieving, it must also be assembled and maintained to mechanically meet the original specification.
At 3000C (and even at 2400C) you need the whole furnace to be designed AND assembled to exacting specs as uneven heating will allow hotspots which may well be fatal to the element. 

These people Oxygon consider themselves to be expert in this field - but I'd hope that whoever supplied your system is equally able to advise. 

Note that in this brochure Oygon talk about " ... such as operating temperatures up to 2800°C (5072°F); ". It seems that at the power levels you were reaching you have been raising the temperatures above the maximum tolerable. Localised melting of the mesh would explain resistance rise. 



Element resistance &amp; resistivity change with temperature.

From the data given I will assume that the values given for V &amp; I allow the element resistance when heated to be calculated.  Any non element portions at such low resistances may affect this assumption adversely.  

Based on the change in resistance per degree K for Tungsten the relative temperature of the element at 4V and 8V MAY be able to be estimated.
delta change per degree K is usually valid over a small delta T compared to T absoliute. This allows linear multiplication of the coefficient by delta T. For large excursions of temperature use of (1+k)^delta_t as a resistance change factor may be more appropriate.

I have used a delta-resistance per degree K of 0.0045 which is the value usually given at 25C. As this procedure is so 'rough' (*or worse :-) ) use of this value should be good enough.

R at 4V x 480A = V/I = 8.333 milliOhms
R at 8V x 390A = 20.5 MilliOhms.
Ratio of resistances = 20.5/8.333 =~ 2.5:1  

Linear

Delta R with temperature per degree K = 0.0045/K
delta T for 2.5:1 change = 2.5/0.0045 =~ 555 C.
Suggested element temperature = 2000 + 555 = 2,555 C.

Exponential

Delta R with temperature per degree K = 0.0045/K  

(1+0.0045)^dT = 2.5
dt = log(2.5) / log(1.0045)
dT = 204C.  

Element temperature = 2000 + 204 ~= 2200C.
Which is inside your spec.

IF
 - the cold temperature of the element is the same at all times, and
 - if the current at and given voltage is the same on multiple runs and
 - if the temperature sensor reports consistent values inside the intended range,
 - Then - all MAY be well.