Flexible demand response (DR) resources can be leveraged to accommodate the stochasticity of some distributed energy resources. This paper develops an online learning approach that continuously estimates price sensitivities of residential DR participants and produces such price signals to the DR participants that ensure a desired level of DR capacity. The proposed learning approach incorporates the dispatch decisions on DR resources into the distributionally robust chance-constrained optimal power flow (OPF) framework. This integration is shown to adequately remunerate DR resources and co-optimize the dispatch of DR and conventional generation resources. The distributionally robust chance-constrained formulation only relies on empirical data acquired over time and makes no restrictive assumptions on the underlying distribution of the demand uncertainty. The distributional robustness also allows for robustifying the otpimal solution against systematically misestimating empirically learned parameters. The effectiveness of the proposed learning approach is shown via numerical experiments. The paper is accompanied by the code and data supplement released for public use.