With the increasing penetration of distributed energy resources such as photovoltaics (PVs), frequent voltage and congestion problems are expected to occur in distribution grids. Existing solution approaches based on centralized or distributed offline optimization cannot handle the dynamics of distribution grids in real time, i.e., by the time the computations are finished, the used data are already outdated as the grid loadings evolve so fast. Moreover, the unfair distribution of costs or burdens among end-users to relieve network issues, for example with PV curtailment, has not been sufficiently considered either. To address these issues, a fairness-incorporated online feedback optimization (OFO) approach is proposed. This approach produces implementable intermediate iterates for inverter-based assets using online optimization and suitably integrated measurements, and thus can work with the distribution grid dynamics. Moreover, it addresses the unfairness issue by modifying the voltage sensitivity matrices and leverages the feedback-based nature of OFO to compensate for the resulting modeling inaccuracies. Case studies based on a 96-bus low-voltage grid demonstrate the effectiveness of the fairness-incorporated OFO to handle the distribution grid dynamics and improve fairness, in both static and dynamic cases, with balanced and unbalanced grid models.