When will ice melt faster? My question is, will ice melt faster if it is left in the water created from melting or will it melt faster if the melted water is drained off?
 A: This will actually depend on both the geometry of your problem and the environment conditions (e.g., is there an "unlimited" reservoir of heat surrounding the ice - in other words, can the environment provide as much heat as needed without cooling down; what is the temperature of the environment).
Energy-wise, it is clear that if the melted water is immediately drained off, the only energy which needs to be supplied is the melting enthalpy for the ice ($333.5 \textrm{ J/g}$ according to wikipedia). If the water stays there, it needs to be heated up before the ice continues melting. The heat capacity of water is ~$4.2\textrm{ J/K*g}$ according to my memory. So, depending on your environment conditions (i.e., environment temperature), the energy needed to warm up the water is only a small additional energy contribution. Still, energy-wise, it is a bad idea to also heat up the melted water if you could also just take it out of your system.
Time-wise, the question is - is the heat transfer from air to ice quicker than the heat transfer from air to water, the heat diffusion throughout the water and the subsequent heat transfer from water to ice? This of course depends on the geometry of the problem (i.e., the respective contact surfaces, the thickness of the water layer) but most of all on the surface transfer rate (if the heat transfer air-ice is quicker than air-water, it will always be good to get rid of the water). Unfortunately, I have problems finding good values for the latter. It might however imply that if you have an unlimited energy supply, it could in some instances actually be a good idea to actually keep the water - I will edit this post if I do find some reliable statement about the transfer speeds.
