87 testarossa starting problem

Aside from the battery issue what you are describing is ignition ONLY when the key is in the "start" position and going dead when the key is in the "run" position. It's a common problem and can be found in numerous threads via the search function. I can't tell you exactly the nature of the problem off the top of my head but there are some simple tests you can do to verify that 12v is present at the correct terminals on the engine.

There should be no damage to the ECU but fuses and relays may be a problem

 

From Steve Magnusson,
If you have a US version TR (F113A040 engine), this is a symptom of either:

1) you've lost the ignition on the 7-12 bank, so the tachometric relay closes during starter motor cranking (key in Pos III), which closes the fuel pump relays, pressurizes the fuel system, and the engine starts -- but the tachometric relay opens once you release the key to Pos II, which opens the fuel pump relays, and the engine dies after the fuel pressure falls, or

2) the tachometric relay is bad and is working OK in mode 1 (key in Pos III), but isn't working in mode 2 (key in Pos II and 7-12 bank ignition firing).

First thing to do is confirm/deny if you have ignition spark on the 7-12 bank during the brief running period. If you search on "TR tachometric relay" you should get some other prior threads on this subject (like this -- http://ferrarichat.com/forum/showthread.php?t=240041). Give a shout if you need help (and please let us know how it works out).