The occurrence of vibrational resonance is investigated in both classical and quantum mechanical Morse oscillators driven by a biharmonic force. The biharmonic force consists of two forces of widely different frequencies $\omega$ and $\Omega$ with $\Omega \gg \omega$. In the damped and biharmonically driven classical Morse oscillator, by applying a theoretical approach, an analytical expression is obtained for the response amplitude at the low-frequency $\omega$. Conditions are identified on the parameters for the occurrence of resonance. The system shows only one resonance and moreover at resonance the response amplitude is $1/d\omega$ where $d$ is the coefficient of linear damping. When the amplitude of the high-frequency force is varied after resonance the response amplitude does not decay to zero but approaches a nonzero limiting value. It is observed that vibrational resonance occurs when the sinusoidal force is replaced by a square-wave force. The occurrence of resonance and antiresonance of transition probability of quantum mechanical Morse oscillator is also reported in the presence of the biharmonic external field.