Ask Mr. Smarty Plants

There are five oaks that are native to Missouri, and we are assuming your oaks are one of these: Quercus alba (white oak), Quercus bicolor (swamp white oak), Quercus macrocarpa (bur oak), Quercus palustris (pin oak) and Quercus rubra (northern red oak). While the acorns and young leaves of oaks are usually mildly toxic, none of these were indicated to be very allelopathic. Allelopathy involves a plant's secretion of biochemical materials into the environment to inhibit germination or growth of surrounding vegetation. Allelopathy enhances tree survival and reproduction. In other words, the plant is protecting itself, its nutrition and water sources, as well as its space, by inhibiting other plants in competition with it. 

We are inclined to believe that the failure of your grasses and perennials are more to be attributed to the shade of the oak, and the fibrous root system. Roots of oaks will often extend far beyond the drip line of the tree itself, and the majority of tree roots are in the upper 6 to 12 inches of the surface of the soil. Also, leaves of deciduous trees can accumulate and smother emerging plants, or promote mold and fungus diseases. Most grasses have difficulty in surviving in that environment, as well as flowering plants; the biggest disadvantage being the amount of shade cast by the mature oak. Basically, you will have to make a choice - big oaks providing shade for the property or flowering perennials? There are shade plants that can be planted under an oak, mostly low groundcovers, but even they have trouble competing.

So, back to the sappy residue you are experiencing in your yard. This sounds a whole lot like aphids, which generate a substance called honeydew, that will drip on lawns, houses, cars and people. It also can cause a sooty mold on the leaves and be very unattractive. This Colorado State University Extension article on Aphids on Shade Trees and Ornamentals will give you some information. They usually are not harmful, and you should avoid pesticides, which are more likely to kill the predators of the aphid than the aphids, themselves.