There is nothing wrong with your schematic that I can see. The gate resistor should only dissipate a few milliwatts average, since it stops conducting once the triac turns on. (10mA for a couple microseconds every 1/2 cycle).

What do you mean "DIP" package for R83- I would expect that to burn up first if it had a similar rating.

I wonder if you have the triac wired incorrectly, or if one of the parts is defective.

Make sure the triac wiring is correct. The view of the package is from the top.

If the triac gate is shorted to the MT1 or the triac is damaged such that it never triggers, you will see something like what you have observed, however R82 will see a lot more power dissipation.

There is nothing wrong with your schematic that I can see. The gate resistor should only dissipate a few milliwatts average, since it stops conducting once the triac turns on. (10mA for a couple microseconds every 1/2 cycle).

What do you mean "DIP" package for R83- I would expect that to burn up first if it had a similar rating.

I wonder if you have the triac wired incorrectly, or if one of the parts is defective.

Make sure the triac wiring is correct. The view of the package is from the top.

If the triac gate is shorted to the MT1 or the triac is damaged such that it never triggers, you will see something like what you have observed, however R82 will see a lot more power dissipation.

If you still need help, posting a photo of the circuit would not hurt.