Carbon-enhanced metal-poor (CEMP) stars are believed to show the chemical imprints of more massive stars (M greater than or similar to 0.8 M-circle dot) that are now extinct. In particular, it is expected that the observed abundance of Li should deviate in these stars from the standard Spite lithium plateau. We study here a sample of 11 metal-poor stars and a double-lined spectroscopic binary with -1.8 <[Fe/H] <-3.3 observed with the Very Large Telescope/UVES spectrograph. Among these 12 metal-poor stars, there are 8 CEMP stars for which we measure or constrain the Li abundance. In contrast to previous arguments, we demonstrate that an appropriate regime of dilution permits the existence of Li-Spite plateau and C-rich stars, whereas some of the Li-depleted and C-rich stars call for an unidentified additional depletion mechanism that cannot be explained by dilution alone. We find evidence that rotation is related to the Li depletion in some CEMP stars. Additionally, we report on a newly recognized double-lined spectroscopic binary star in our sample. For this star, we develop a new technique from which estimates of stellar parameters and luminosity ratios can be derived based on a high-resolution spectrum alone, without the need for input from evolutionary models.