We report the dependence of the oxygen isotope shift on transition temperature ${\mathit{T}}_{\mathit{c}}$ in the system ${\mathrm{YBa}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{La}}_{\mathit{x}}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{\mathit{z}}$ with 0\ensuremath{\le}x\ensuremath{\le}0.5. We find a significant oxygen isotope shift at low temperatures (${\mathrm{\ensuremath{\alpha}}}_{0}$=0.38 at ${\mathit{T}}_{\mathit{c}}$=38.3 K), which decreases gradually with increasing ${\mathit{T}}_{\mathit{c}}$ and finally falls rapidly above 73 K to ${\mathrm{\ensuremath{\alpha}}}_{0}$=0.025 for ${\mathit{T}}_{\mathit{c}}$=92.3 K. Our data are well fitted by the expression ${\mathrm{\ensuremath{\alpha}}}_{0}$(${\mathit{T}}_{\mathit{c}}$)=0.5(1-${\mathit{T}}_{\mathit{c}}$/${\mathit{T}}_{\mathit{c}0}$${)}^{0.59}$, with ${\mathit{T}}_{\mathit{c}0}$=92.8 K. Our results suggest a dominant role for conventional electron-phonon coupling in the high-temperature cuprate superconductors, with strong suppression of the isotope effect as ${\mathit{T}}_{\mathit{c}}$ approaches its maximum value in this system.