When two bodies at different temperatures are placed in contact, heat flows from a body at higher temperature to a body at lower temperature until they reach to the same temperature which is called thermal equilibrium condition.

We a hot tea is touched the finger acts as a cold body and the hot tea cup acts as a hot body. The molecules of hot tea are vibrating with greater average kinetic energy than that of finger.

When in contact molecules of hot tea lose their energy to molecules of the finger until they vibrate with same average kinetic energy(temperature). In this way tea loses its heat to finger, hence on touching it feels hot.

Similarly, when touching an ice cube, there is loss and gain in kinetic energy of molecules of finger and ice until they reach the same average kinetic energy(temperature). In this case ice is cooler and the finger loses its heat to ice. 

Hence, the observer's finger is cooled on touching ice.