For 3,4-dimethoxybenzyl-protected alcohols, Greene suggests DDQ (as DavidW wrote), and the overall mechanism is oxidative (you get your alcohol + 3,4-dimethoxybenzaldehyde). The detailed mechanism is probably via free-radicals, as you have a quinone around.
For 3,4-dimethoxybenzyl-protected amines, Greene gives "H2SO4/anhydrous TFA, anisole". I think it means either acid + anisole. In this case, the mechanism is probably via carbocation, i.e. you get N-protonation followed by departure of 3,4-dimethoxybenzyl cation, which is scavenged by anisole. Or maybe it's anisole which attacks the N-protonated substrate in the transition state.