My big tree I have in my backyard has started dropping lots of sap, not just from the trunk but from all over the tree.
Eucalyptus Tree Dropping Sap
Certified GKH Gardening Expert
Your tree could be infected with a pest called Red gum lurp Psyllid.
The dripping of sap is actually called honeydew, and it is the secretions from the insects.
Here are some links with more information and treatment action.
If you tree is large you may need a professional to treat the tree.