Cavity quantum electrodynamics (cavity QED) with neutral atoms is a promising platform for quantum information processing and optical fiber Fabry-Pérot microcavity with small mode volume is an important integrant for the large light-matter coupling strength. To transport cold atoms to the microcavity, a high-power optical dipole trap (ODT) beam perpendicular to the cavity axis is commonly used. However, the overlap between the ODT beam and the cavity mirrors causes thermal effects inducing a large cavity shift at the locking wavelength and a differential cavity shift at the probe wavelength which disturbs the cavity resonance. Here, we develop a feedback and compensation scheme to maintain the optical fiber microcavity resonant with the lasers at the locking and probe wavelengths simultaneously. The large cavity shift of 210 times the cavity linewidth, which makes the conventional PID scheme ineffective can be suppressed actively by a PIID feedback scheme with an additional I parameter. Differential cavity shift at the probe wavelength can be understood from the photothermal refraction and thermal expansion effects on the mirror coatings and be passively compensated by changing the frequency of the locking laser. A further normal-mode splitting measurement demonstrates the strong coupling between 85Rb atoms and cavity mode after the thermal effects are suppressed, which also confirms successful delivery and trapping of atoms into the optical cavity. This scheme can solve the thermal effects of the high-power ODT beam and will be helpful to cavity QED experimental research.