A recirculation pump can be beneficial when using a tankless water heater, especially in larger homes where the distance between the heater and the fixtures is significant. The pump keeps hot water readily available at the tap, reducing wait times and water waste. However, it's true that it can somewhat counteract the energy efficiency of a tankless system, as it requires energy to operate.
Locating the tankless water heater closer to the bathroom or kitchen can indeed improve efficiency. Ideally, you want the heater to be as close as possible to the areas where hot water will be used most frequently. This can minimize lag time and water waste, making the system more efficient.
In terms of cost, a recirculation pump can range from $300 to $600 for the unit itself, plus installation costs, which can vary widely by region. If you're considering this upgrade, I recommend getting 2-3 local quotes through Handyman.com to find the best option for your home.
— Assisted answer