Searches are presented for narrow heavy resonances decaying to a top quark and a bottom quark using data collected by the CMS experiment at a center-of-mass energy of 13 TeV. Final states that include a single lepton (electron, muon), multiple jets, and missing transverse momentum are analyzed. No evidence is found for the production of a W' boson, and the production of right-handed W' bosons is excluded at 95% confidence level for masses up to 3.6 TeV depending on the scenario considered. Exclusion limits for W' bosons are also presented as a function of their coupling strength to left- and right-handed fermions. These limits on a W' boson decaying via a top and a bottom quark are the most stringent published to date. Projections for future searches with an integrated luminosity of up to 3 inverse attobarns are also presented, and suggest that W' boson masses above 4 TeV could be excluded.