Every circuit component always has some small property of every other component like resistance or capacitance or inductance. However, one thing I noticed is that it's possible for a spontaneous change in the current to overheat a part of a circuit, and it would seem this would be due to the component's natural tendency to resist changes in current, aka inductance. At what point is this a problem in a circuit, and how can it be overcome?
However, one thing I noticed is that it's possible for a spontaneous change in the current to overheat a part of a circuit, and it would seem this would be due to the component's natural tendency to resist changes in current, aka inductance.
Your premise is flawed. When a part has parasitic inductance, it doesn't cause the part to heat up.
When a part has parasitic inductance, it's because in order for current to flow through the part, a magnetic field must be built up around the part or the wires within it. Creating this magnetic field requires energy, which means a voltage must be applied to the part.
But this energy is not converted to heat. It's stored in the magnetic field, and when the current is reduced or reversed, the energy will be returned to the circut.
As to what actually causes the heating you observed, you'll have to give a more specific example or explanation of the circuit you're working with, because generally it's not common for a circuit to produce heat as a result of changes in current, although I can imagine it's possible to make a circuit to do that if you worked at it.
Heating or Thermal energy dissipated is a result of I x I x R x t. Reducing the internal resistance of the particular component (if you can), the current flow or the time of current flow will reduce heating. Inductance has nothing to do with heating as @Phantom pointed out.
To answer the question in the title : yes it can be a problem and it must be recognised / solved / corrected in the design so the device / circuit does what it is designed for.