I'm not engineer, but here are a few comments.
1) MMM did this to heat his house, but put the coils in the floor:
http://www.mrmoneymustache.com/2014/02/16/the-radiant-heat-experiment/2) If you think this is free, I have some bad news. As the heat transfers from the water, through the copper pipe, and warms the air - the water gets cold (or more precisely: loses its heat). So when it goes back to the hot water heater, it cools the average temperature of the water in the tank and the water heater needs to kick on to heat up the water again.
3) Unless you have a fan or some other method of moving the heated air throughout the room, I would expect that you wouldn't get distribution of the heated air. Sure, some will move around as the nature process of air trying to rise, etc., but it won't heat a regular sized room, unless you really have a lot of pipe and it is over quite an area.
4) My personal experience with this type of set up is that it's basically a heated towel rack - they're pretty popular in Europe:
https://www.cleanandtoasty.com/7-reasons-why-towel-warmers-are-so-popular-in-europe-3-will-surprise-you/. Great for heating your towel and slightly warming a really small bathroom, but not really feasible for heating a whole room .... and no one claims that it is free (doesn't use additional energy).
But it's always good to be thinking about alternative methods for energy use/conservation.