We comprehensively study the charged Higgs boson contributions to the leptonic Bq−→ℓν¯ (q=u, c) and semileptonic B¯→Xqℓν¯ (Xu=π, ρ; Xc=D, D*) decays in the type-III two-Higgs doublet model. We employ the Cheng-Sher ansatz to suppress the tree-level flavor-changing neutral currents in the quark sector. When the strict constraints from the ΔB=2, b→sγ, and pp(bb¯)→H/A→τ+τ− processes are considered, parameters χtqu from the quark couplings and χℓℓ from the lepton couplings dictate the leptonic and semileptonic B decays. It is found that, when the measured Bu−→τν¯ and indirect bound of Bc−→τν¯ obtained by LEP1 data are taken into account, R(D) and R(π) can have broadly allowed ranges; however, the values of R(ρ) and R(D*) are limited to approximately the standard model (SM) results. We also find that the same behaviors also occur in the τ-lepton polarizations and forward-backward asymmetries (AFBXq,τ) of the semileptonic decays, with the exception of AFBD*,τ, for which the deviation from the SM due to the charged Higgs boson effect is still sizable. In addition, the q2-dependent AFBπ,τ and AFBD,τ can be very sensitive to the charged Higgs boson effects and have completely different shapes from the SM.