A novel method for high-sensitive measurements of optical attenuation in metal-coated optical fibers in a wide range of wavelengths is demonstrated. Some part of the radiation transmitted through the waveguiding core of the metallized silica fiber is scattered and eventually absorbed in the carbon layer or the metal coating, thus heating it. Absorption in the silica core also contributes to the overall fiber heating. The method used for determination of optical attenuation coefficients of metallized fibers is based on the measurement of the change of temperature-dependent electrical resistance of metal coating induced by transmitted laser radiation. A number of single-mode and multimode metallized fibers with different geometry were investigated using several laser sources operating in visible and near infrared ranges. Experimentally obtained spectral dependence of optical losses of copper-coated fibers was analyzed. Possible explanations for optical attenuation mechanisms at different wavelengths and in different fibers were proposed. The obtained results can help to optimize various devices based on metal-coated fibers, such as laser radiation power fiber sensors or high-power laser sources.