What uses less energy overall, running how water from the sink and then boiling that or starting with the water cold?

Well, I'm assuming here that what you're talking about is something like a kettle or saucepan where you're trying to boil water? If this is the case, then boiling cold water is likely to be vastly more efficient. Here's my reasoning for that ...

When you talk about 'hot water from the tap' what you're probably talking about is water that has been heated in a tank in the 'hot press' using a gas boiler or electric heating elements. This is a pretty large volume of water so you need a lot of energy to raise its temperature simply due to that volume. It's why many of those systems require an hour or more before you've enough hot water for a shower or bath.

So you heat that large volume of water, wait until the temperature is high enough to be useful, then transfer it to a kettle or saucepan to boil it. What you're doing in this situation is drawing a relatively small amount of the large volume you're after heating, and then using the kettle or saucepan to heat it more. This isn't a very efficient way of doing it.

The more efficient way is to draw a small amount (a kettle's worth or saucepan's worth) and heat it. It may start off cold, but you've far less volume of water to heat.

Having said that though, the most efficient way would be if you've heated water for a bath or shower, have that bath, then have hot water left over - you draw this water for the kettle or saucepan and boil it since it's already hot and would just cool in the tank otherwise.

It depends on several things: How efficient is you hot water heater? How far is it between the hot water heater and the tap? How would you boil the cold water?

In general, microwaving is the most efficient way to heat water, so if you use a microwave, it's more efficient to start with cold water.

If you heat water on a stove, it's also more efficient to start with cold water if your hot water heater and your stove use the same source of energy (gas or electricity), since you lose some heat to the pipes when you draw hot water from the tap.

If your stove and hot water heater use different energy sources, it's hard to say.