You live on a third floor with your hot water storage tank in the basement Where would a tankless heater need to be located to work effectively?

A tankless and a gas(with a normal tank) water heater both burn gas to heat the water, so they need vents to bring air into the space where they are located, and a venting system for the exhaust. Some considerations for tankless: 1) they require larger gas supply pipes; 2) they can be placed on most walls, but you still need room around them; and 3) they heat the water as it flows through the pipe, so you still have to deal with the cold water which is in the pipe. I mention the last one because many people think that they will immediatley obtain hot water.

For your situation, placing the tankless heater on the third floor would require new gas pipes. Commonly gas pipes are 3/4" in a home, and a tankless unit may need 1" at a minimum. Having the water heater on the same level would provide you with hot water sooner. Placing the tankless unit where the hot water storage tank is may require less Plumbing, so it could be an easier task. The water will take longer to heat, and you could be wasting energy of the heated water in the pipe. This can be partially resolved by insulating the hot water pipes. Insulating these pipes is a wise idea for efficiency in either location.