Planning paths that maximize information gain for robotic platforms has wide-ranging applications and significant potential impact. To effectively adapt to real-time data collection, informative path planning must be computed online and be responsive to new observations. In this work, we present IA-TIGRIS (Incremental and Adaptive Tree-based Information Gathering Using Informed Sampling), which is an incremental and adaptive sampling-based informative path planner designed for real-time onboard execution. Our approach leverages past planning efforts through incremental refinement while continuously adapting to updated belief maps. We additionally present detailed implementation and optimization insights to facilitate real-world deployment, along with an array of reward functions tailored to specific missions and behaviors. Extensive simulation results demonstrate IA-TIGRIS generates higher-quality paths compared to baseline methods. We validate our planner on two distinct hardware platforms: a hexarotor unmanned aerial vehicle (UAV) and a fixed-wing UAV, each having different motion models and configuration spaces. Our results show up to a 38% improvement in information gain compared to baseline methods, highlighting the planner's potential for deployment in real-world applications. Project website: ia-tigris.github.io.