Do not run that trimmer without oil in the petrol, or it will seize up and have to be thrown away. Very nearly all trimmers are crankcase induction 2 strokes, and the only lubrication the engines get is from the oil in the fuel. (Honda sells 4 stroke trimmers that use different lubrication arrangements, but I've never heard of Homelite doing so, and in any case you said it is a 2 stroke.)
If you are sure it is the oil in the fuel that is making it refuse to start (this would be very unusual, by the way) it sounds as if the carburetor is set a bit lean. The oil increases the viscosity of the fuel a bit and also tends to coat the inside of the jet, which may cause the air/fuel ratio to be a little leaner when the oil is present. You need to go through the carburetor manufacturer's procedure for adjusting the mixture. If you tell us what brand and model of carburetor you have, we can direct you to the carburetor manual. I suggest you do not attempt to adjust the mixture without having this information, because you are likely to make things worse.
