Harmonic potentials provide globally convergent potential fields that are provably free of local minima. Due to its analytical format, it is particularly suitable for generating safe and reliable robot navigation policies. However, for complex environments that consist of a large number of overlapping non-sphere obstacles, the computation of associated transformation functions can be tedious. This becomes more apparent when: (i) the workspace is initially unknown and the underlying potential fields are updated constantly as the robot explores it; (ii) the high-level mission consists of sequential navigation tasks among numerous regions, requiring the robot to switch between different potentials. Thus, this work proposes an efficient and automated scheme to construct harmonic potentials incrementally online as guided by the task automaton. A novel two-layer harmonic tree (HT) structure is introduced that facilitates the hybrid combination of oriented search algorithms for task planning and harmonic-based navigation controllers for non-holonomic robots. Both layers are adapted efficiently and jointly during online execution to reflect the actual feasibility and cost of navigation within the updated workspace. Global safety and convergence are ensured both for the high-level task plan and the low-level robot trajectory. Known issues such as oscillation or long-detours for purely potential-based methods and sharp-turns or high computation complexity for purely search-based methods are prevented. Extensive numerical simulation and hardware experiments are conducted against several strong baselines.