Peach trees produce abundant, sticky sap that can ooze out of the tree for a variety of reasons. The sap hardens into gummy yellow to orange nodules on the bark.Sap oozing can be caused by insects, diseases, injury or a natural response to environmental conditions in dwarf peaches.each twig borers and shothole borers make small holes with limited sap oozing. Peachtree borers can kill a tree by boring into the lower trunk. Dig them out of the tree with a pocketknife. Take care not to remove too much wood or you will kill the tree.