Aims: We aim, on the one hand, to study the possible differences of Li abundances between planet hosts and stars without detected planets at effective temperatures hotter than the Sun, and on the other hand, to explore the Li dip and the evolution of Li at high metallicities. Methods: We present lithium abundances for 353 main sequence stars with and without planets in the Teff range 5900-7200 K. We observed 265 stars of our sample with HARPS spectrograph during different planets search programs. We observed the remaining targets with a variety of high-resolution spectrographs. The abundances are derived by a standard local thermodynamic equilibrium analysis using spectral synthesis with the code MOOG and a grid of Kurucz ATLAS9 atmospheres. Results: We find that hot jupiter host stars within the Teff range 5900-6300 K show lower Li abundances, by 0.14 dex, than stars without detected planets. This offset has a significance at the level 7σ, pointing to a stronger effect of planet formation on Li abundances when the planets are more massive and migrate close to the star. However, we also find that the average vsini of (a fraction of) stars with hot jupiters is higher on average than for single stars in the same Teff region, suggesting that rotational-induced mixing (and not the presence of planets) might be the cause for a greater depletion of Li. We confirm that the mass-metallicity dependence of the Li dip is extended towards [Fe/H] ~ 0.4 dex (beginning at [Fe/H] ~-0.4 dex for our stars) and that probably reflects the mass-metallicity correlation of stars of the same Teff on the main sequence. We find that for the youngest stars (<1.5 Gyr) around the Li dip, the depletion of Li increases with vsini values, as proposed by rotationally-induced depletion models. This suggests that the Li dip consists of fast rotators at young ages whereas the most Li-depleted old stars show lower rotation rates (probably caused by the spin-down during their long lifes). We have also explored the Li evolution with [Fe/H] taking advantage of the metal-rich stars included in our sample. We find that Li abundance reaches its maximum around solar metallicity, but decreases in the most metal-rich stars, as predicted by some models of Li Galactic production. Based on observations collected at the La Silla Observatory, ESO (Chile), with the HARPS spectrograph at the 3.6 m ESO telescope, with CORALIE spectrograph at the 1.2 m Euler Swiss telescope and with the FEROS spectrograph at the 1.52 m ESO telescope; at the Paranal Observatory, ESO (Chile), using the UVES spectrograph at the VLT/UT2 Kueyen telescope, and with the FIES and SARG spectrographs at the 2.5 m NOT and the 3.6 m TNG, respectively, both at La Palma (Canary Islands, Spain).Tables 3-6 are available in electronic form at http://www.aanda.org