+Has it happened before that resetting the PMU will resolve starting up to a blank screen?+
I'm not sure about that, but I do know that it's the first thing AppleCare had me try when I called in about mine. It won't hurt, and it may help.
+If doing so does work, do I still need to have the logic board replaced?+
That's hard to say, since it is not uncommon for it to work sporadically with a bad logic board. I would try it, and if it works, use it as long as possible.
Does squeezing the case to the left of the trackpad allow you to start up after being shut down overnight? If so, you may want to consider trying the shim fix suggested by John Sawyer in the Apple Discussions iBook G3 (Dual USB) Displays Forum:
[Y]ou can often fix this yourself by removing the iBook's bottom housing, and placing a shim of any sort, about 1mm to 1.5mm thick, onto the raised square on the bottom shield. I use a Scotch mounting square--you can get them in hardware stores and many grocery stores. When you reinstall the bottom case, it will press against this shim, which will press against the graphics chip, and may allow the chip to come into better contact with the logic board.
John Sawyer
CJS Macintosh Repair
Instructions for removing the lower case:
iFixit.com
Do you have access to another Mac with FireWire? If so, you may be able to get the iBook to remain responsive in
FireWire Target Disk Mode so you can retrieve your data by this method:
If the iBook is on, shut it down and wait 30 seconds (or longer). Pick it up in your left palm to the left of the trackpad and squeeze the case there between your palm and left thumb as you push the power button and key combination to start up in FireWire Target Disk Mode. Do not let up the pressure on the case. Continue applying this pressure as you save your data to the other Mac. If you let up and the iBook dies or freezes, you will have to start over, allowing it to rest several hours or overnight before beginning again.
Good luck!