I can't see any form of bug on it but I can see some sooty colouring on some of the leaves. My roses have some of this on them but they arenot dying.
The Sooty Mould is produced when a sap sucking insect such as scale or mealy bug exudes a sugary substance. The Sooty Mould grows on the sugary excretion. Once the insect pest is controlled then the Sooty Mould will no longer develop. Have you noticed any small wax like lumps on the branches or twigs? Apply some PestOil to control the scale. You may need more than 1 spray to control this pest. I have attached the link to the pictures on the Yates Website to help identify the Pests. Lilly Pillies are rainforest plants and do require regular watering. Check the soil and see whether it is moist below the surface.
Answered: 5 Sep, 2012