This same thing happened to me.
I got a new disc drive, took the board out from the old one and added it to the new drive and all that, put everything back together and start it up…and it makes a faint power on sound but doesn't start up.
I tore down my Xbox all the way down to the motherboard and examined all the contacts and nothing looked wrong. I also thought maybe I damaged the front ribbon cable, but then I tried to jump the pins of the ribbon cable socket and it was the same result…a faint beeping sound.
I decided to try to turn it on with just the motherboard and the fan connected…and it powered on perfectly. I then decided to power it on with just the hard drive connected…it powered on perfectly again. I then plugged in the disc drive and tried to power it on….faint beeping but no power.
So I realized it had to be the disc drive…I opened up the disc drive to discover that like an idiot I hadn't attached all the cables! I attached the one I didn't have attached and all is working now.
TL;DR - Check the connections inside the disc drive! Maybe this is your problem. Hopefully this helps people who have this same problem down the line.