Two potential problems:
1.) The heater core might reduce the water pressure to the rest of the house
2.) It also might clog up eventually if your water has a high mineral content. A water softener might help. You'll probably need to run CLR through it every few years.
It might probably be better to have something like this, although this doesn't help to reduce the water heater bill:
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However, the original idea would probably look something like this:
$25 - New car heater coil - from auto parts store - need to test if it can hold the pressure
$20 - PEX or Sharkbite plumbing connectors - from hardware store
$10 - PEX plumbing tubing - from hardware store
Optional:
$150 - Automatic Water Shut off Valve - these are normally used next to a water heater in a house to shut off the water supply if the water heater leaks. You could justify this cost by having the heater core and water heater next to each other so that the heater coil is helping a bit to heat the water heater and the auto shutoff valve is protecting you both from a heater coil failure and water heater failure
$20 - Materials to direct heat into heater core. This might be done using a "hot/cold aisle" layout that's used in server rooms, maybe using drapes or something similar to
http://www.neuralenergy.info/2015/09/green-data-centers.htmlOne thing you would need to test well is that the connection from the water supply to the heater core is very secure. It's a 3/4" pipe coming from the main shutoff valve and you'd probably want to have a soldered adapter to 3/4" on both ends of the heater core to make sure it doesn't leak.