We theoretically examine the Fano lineshapes of silicon-based compound microring resonators consisting of a single resonator channel dropping filter linked to a loop as a feedback structure. All possible optical effects for the continuous-wave operating regime, such as linear absorption or scattering, two-photon absorption, free-carrier absorption and dispersion, thermo-optics, are simultaneously considered. We show that sharp Fano resonances can be tuned by variation in the coupling coefficients, length of feedback loop, effective free carrier lifetime and the temperature inside the device. Tunable Fano lineshapes open up opportunities for applications in sensing, computing, and communications.