What is the value of this resistor?
Based on what's visible in this video, the resistor (R1) appears to be 1k Ω ±5% (~0.5 W?).  Here are some screenshots:

Front of board.  A bit blurry but the best shot I could find of the resistor.

Back of board.  Resistor is across the two right-most pads on the lower edge.
And here's a schematic:



simulate this circuit &ndash; Schematic created using CircuitLab
Schematic is laid out to match part locations on the board as viewed from the top.  R1 is the burned-out resistor that the question is asking about.  It's not very visible but the OP has verified that the part labelled D5 is a DIAC (part marked DB3).
Here's the schematic redrawn a bit more clearly:



simulate this circuit
Use of a DIAC on the gate of a TRIAC is discussed here.  The datasheet for the DIAC can be found here (thanks to @PaulGhobril for finding that).  The datasheet for the TRIAC can be found here; the part used on this board does not require a snubber.
My guess is that in addition to R1, at a minimum C3 is probably also toast (shorted).