R15 is 100k ¼W (Brown Black Yellow) and is located next to Relay RL2. It is prone to overheating and you might replace it with 100k ½W.
While you have it apart replace the electrolytic capacitors C4 & C7 both 22uF 63V and C6 4.7uF at 63V. Be sure to observe the polarity of the capacitors and get them the right way round ±.
These capacitors are prone to degradation due to being in an over warm environment all the time. Electrolytic capacitors can be badly affected by heat. Try to get the same or higher voltage rating (but not lower) and 105 ºC temperature rating for longer reliability/life or at least 85 ºC.
An article here tells all about it - with expanding pictures.
http://kaijaks.co.uk/blog/nick/2008/02/10/dodgy-old-boiler
I believe that R15 is placed across the gas valve as a bleed resistor to help the valve close more quickly - as such it takes current as long as the gas valve is open - a burnt R15 is a feature of this PCB - they all burn in time and is nothing much to worry about. I think the boiler will work even without R15 in circuit - the gas valve will just close more slowly
All these components are available very cheaply from a hobby electronics shop like Maplins in the UK.
Good luck...
April 2015