What would happen if the output leads of a multi-phase transformer were shorted together?
Regardless of use / topology 


Joining together wires
that should remain asunder
will may well result in sparks
and could result in thunder. 


It is unlikely that the inverter is a 3 phase one.
 It is very likely that it is single phase.  

It would be helpful if you stated the number of tubes and wires more clearly - what you have said could mean several things (or my brain is not working properly)(always possible).

You say "each pair of tubes".
 This implies at least 4 tubes, with them being arranged ion pairs.

You say "3 pairs of wires" impling (or in fact indicating) 6 wires. 

"3 phase HV for each pair of tubes requires at least 1.5 wires per tube - but you could mean 3 wires per tube, or two, or ... .

HOWEVER

Shorting together 2 wires from connected windings on a CFL would indeed result in heating up - usually with destruction of one of the wires and/or death of the whole CFL, probably in the 0.02 to 5 second range (0.016 to 5 seconds in USA and selected other places).

It is possible that they are using filaments in the tubes - in which case a pair of wires would both convey filament low voltage and act as a path for the high voltage to one end of the tube.

It is also possible that the two wires in a pair go to opposite ends of the same tube and provide the HV excitation to the tube.

Regardless of use / topology - joining wires together that should remain asunder  may result in sparks and, just possibly, thunder. 



It's been a while since I pulled any dead CFLs apart.
 I just went and had a dig through my daed CFL collection (every workshop should have one) to refresh my memory re typical practice.

A CFL is essentially a conventional fluorescent tube writ small and folded, with an electronic "ballast" to start the tube with high voltage and then provide the correct (much lower) voltage required to operate it at its rated power. 

I dismantled a coild tube bulb - which is essentially a single 2 ended glass tube, and a 3 straight loop tube which I expected to consist of 3 separate loops. It didn't. The loops are combined by bridging tubes so that it only has two electrical end amongst its 6 apparent mechanical ends - it's a single tube for the internal 'gas', and electrically. 

In my case, and probably in most tubes that are exopected to have good lifetimes, there is a heater/filament at each electrical end of the tube and HV is fed to each end along the heater wire. So, each 2 ended tube has 2 wires per end.
 shorting these two wires together would stop the heater heating and would quite possibly create an informal external heater using the usually rather fine wiring. 

Shorting two wires together from each end of a tube would result in the inverter / ballast having an apoplectic fit, which would quite possibly be fatal for it. worst case it could also be fatal for the person doing the shorting if power was applied at the time. 



Health &amp; Safety notice:

Note that CFL tubes contain a small amount of mercury. If you don't eat the tubes after use and avoid breaking them while thy are operating and breathing the fumes and if you wash your hands after breaking one, while all the while refraining from licking your fingers, then odds are that the mercury amount is too small to matter very much. YMMV . 

Cutting yourself on broken-CFL glass may be a worse than average idea. As well as the mercury there is phosphor which is not intended for internal use.