Why do people bury chili underground?
Typically, dutch ovens and other heavy, especially cast iron pots would be buried to allow a slow dissipation of the heat.  This will allow the contents to continue to cook at a very low and slow pace, usually to allow the release and melding of more flavor.  In the case of chili for instance, this could allow more of a dried chili essence to reconstitute and become more uniform throughout the pot.  The insulation of the ground will slow the cooling and allow this process to extend much longer than air would.  Additionally, packing soil around the pot would tend to act as a barrier to hold in more of the aromatics than in air, combined with the slower cooling to again allow more absorption and a more even distribution for a more mellow taste to the dish.

Especially pots such as dutch ovens, this process is often done without additional heat, or in ground warmed by a fire, or with added coals or hot rocks depending on how long you want this process to continue and if you want more actual cooking to occur.

Fully duplicating this process inside would be difficult, but a somewhat close approximation would be to heat up an oven.  Put in your covered pot, preferably tightly covered.  Close the oven then turn it off or down to a very low setting, and keep it closed for a few hours and you should get much of the same effect.  I would call this process marrying or melding the flavor rather than saving it.