Is the 100 ohm resistor needed in this mains power supply?
Yes, you need it.

If you don't have it, some part of your circuit will burn out and be destroyed.  Which part will actually go "poof" or "bang" first can't really be predicted.



You've been told before, but I'm going to tell you again:

That circuit can kill you.

You must operate that thing in an enclosed box.  

Do not allow any user to touch any electrical part of that thing when it is operating.

If you connect the output of that thing to another device, then that device can now kill you, and should be enclosed when operating.

You must assume all parts of that thing to be at line voltage.

You should find a safer powersupply circuit to build.