Lignin is an attractive feedstock for low molecular weight biobased phenols using depolymerization strategies such as reductive catalytic fractionation or fast pyrolysis. Such strategies often yield a product mixture enriched in lignin-derived monomers with methoxy substituents. Selective catalytic hydrodeoxygenation (HDO) is an effective methodology to demethoxylate these monomers into valuable alkylated phenols. Here, we report the use of non-precious Cu based catalysts supported on SiO2, ZrO2, TiO2 (various forms), MoO3-ZrO2, and MoO3-TiO2 in a continuous fixed-bed reactor at elevated temperature and pressure (300–360 °C, 10 bar) for the selective demethoxylation of guaiacol. Among the various catalysts, Cu/TiO2-P25 was found to be an effective and highly stable catalyst (100 h on stream) with a selectivity of 87% to demethoxylated compounds like phenol and cresols at a guaiacol conversion of 98%. A correlation was found between the oxygen storage capacity of the support (TiO2-P25, TiO2-A HSA, TiO2-R, and MoO3-TiO2) and guaiacol conversion, indicating that this property plays a role in the catalytic cycle. Besides, the demethoxylation of 4-n-propylguaiacol and a realistic guaiacol-rich feed isolated from a representative pyrolysis oil was successfully demonstrated. 87% of the guaiacols present in the feed were converted to demethoxylated phenols with a selectivity of 81%.