SHARP Projector turned off and will not power back up

It could just be the bulb, when you replace the bulb, handle the bulb with cardboard, and avoid touching the bulb with your fingers. The oil on the bulb, from your fingers, can make the bulb overheat.
it could be the power supply...
If you are handy replacing capacitors and MOSFETs, you CAN probably get it running, but for the average Joe,

The power supplies in these projectors burn out, mostly the MOSFETs and capacitors. And the projector bulb burns out,
the plastic mirrors inside also warp and crack.
Be prepared to deal with this scenario if you want to fix an LCD projector.
Not really reliable, very expensive to repair, better off buying a flatscreen TV.
OK if I repair it myself, not super expensive, maybe under $100?
Overall, I would call LCD projectors a poor investment. There are CRT video projectors that will be much more reliable, but they cost a lot.
If you NEED a projector that is reliable, buy a CRT projector. It's not a good investment to sink money into an LCD projector, or to spend money getting one repaired. LCD projectors could certainly spend a lot of time in the repair shop.

My guess is it has something to do with the power supply as even if the bulb was burnt out the projector should still power up (or at least acknowledge it's plugged it by lighting up the red LED). However, nothing happens when it's plugged in.

The last one I repaired was over $300. It really was not worth it, compared to just buying a flat screen TV.
Blown MOSFETS, shorted capacitors, warped and cracked plastic mirror. After that it worked again.

Darn. Not what I wanted to hear :-(

Maybe I can just sell it for parts? The bulb should still have life left on it.

However, nothing happens when it's plugged in.