MacBook Pro is not turning on monitor
It could very well be a broken logic board.  But try to do the following and see if it fixes it :

Reset System Management Controller as per these instructions

Reset PVRAM as per these instructions

Doing these two resets generally fixes most hardware based issues on Macs.  You can also eliminate that its your monitor by trying to see if you can replicate this problem on other screens.

But I would trust the advice you've been given by Apple on this one.