keep burning up resistor 3, which in the diagram is a 56ohm resistor.
Well a quick sum shows that the resistor is burning 8.8W worst case so it is no wonder a quarter watt resistor sizzled.
You are dealing with 120V mains so that gives a peak voltage of 170v. The total resistance across this is 56 + 330 = 385 R, that with 170v across it gives about 0.4A so you have a peak of 53W on the other resistor.
Of course you don't get this continuously because the triac should fire and reduce the voltage to zero but is is an idea of the size of things.
The problem is that the circuit you used is not very suitable for dimming, it is more use as a simple switch. Yes you can make it dim if you synchronise the firing of the triac to the mains but it is a lot of trouble. There are SSRs that give you simple dimming options with PWM. These work by the PWM changing the brightness of an LED (in an opto isolator) and that being used to reduce the time constant of an RC circuit in the gate of the triac fed through a diac (DIAC - Wikipedia ). This gives you simple phase modulated switching which is what you need for dimming. This is a simple example of a circuit using a pot to control the dimming:-
This is the sort of waveform you are trying to get:-
I am also concerned that the optoisolator I have is for 240v and not 120v, but I had assumed (again) that it could run at either voltage,
That is no problem you can run it at any voltage.