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Step 1: Understanding the Effect of Ice Melting
When a piece of ice floats in water, it displaces a volume of water equal to the weight of the ice. This displacement is in accordance with Archimedes' principle.

Step 2: Ice Melting in Water
As the ice melts, it turns into water. Since the ice has already displaced the same volume of water as its weight, the volume of water produced by the melting ice will exactly occupy the volume that was displaced. Thus, the water level will remain unchanged and no water will overflow.

Step 3: Analysis for Denser Liquid
If the beaker is filled with a liquid denser than water, the floating ice will displace less volume of this denser liquid compared to the volume it displaces in water. When the ice melts, the denser liquid will not overflow because the smaller volume displaced will equal the volume of water produced from the melting ice. Therefore, no overflow occurs.

Step 4: Analysis for Less Dense Liquid
If the beaker is filled with a liquid that is less dense than water, the ice will displace more volume of this less dense liquid. When the ice melts, the volume of water produced from the melting ice will occupy less space than what was displaced initially. Hence, this scenario may cause overflow since the volume of liquid displaced becomes less than the volume of the ice plus the water formed from melting.

Therefore, the final answer based on each scenario is as follows:
1. Ice melts in water: No overflow
2. Ice melts in a denser liquid: No overflow
3. Ice melts in a less dense liquid: Overflow likely occurs.