Triac Circuit Won't Fire With Small Loads
Triacs have what's called a 'Holding current'. This is the current required to ensure the Triac will remain on for the rest of the half cycle. Look at the datasheet for your devices, but I assume the devices you have do not have the required holding current.
You don't show your schematic, but I assume it's a MOC3020 driving a power Triac.
With this type of circuit, at very low load currents only the MOC3020 turns on (there is insufficient gate current to turn on the power Triac.   

As the load current increases you reach a threshold where the MOC3020 current is enough to turn on the power Triac (which immediately turns off the MOC3020). Now, if you have enough current to keep the power Triac on all is good....but if you don't have enough current then it turns off. 

This is a common problem when implementing zero crossing turn on, but not normally an issue if you drive the MOC3020 LED continuously (asynchronously).