The effects of H2o2 (hydrogen peroxide) with oils I have some 35% $\ce{H2O2}$ that I've been cutting down to 3% to create a product. Cheaper than buying pre cut solution from store.

However, I've been thinking about trying to try it out with producing a new product which would induce mixing of the $\ce{H2O2}$ with pure oils and essential oils. I am not sure if the $\ce{H2O2}$ would break down the oils and render them useless in the product or not.

I select $\ce{H2O2}$ cause I want it to be able to kill germs to help kill a particular smell. But I want the oils to maintain their properties in which they keep there defined smells. So pretty much the $\ce{H2O2}$ is a catalyst to make to oils more liquidity able to spray and also helping hide original scent.
Oils will oxidize (and burn, too). With dilute H2O2, it's not a rapid reaction (in laymans terms)... some oils (like linseed) will oxidize more rapidly than others (like mineral oil). 

Essential oils are not really oils... they are more like phenolic and terpenic extracts.