Why does a steel rod keep increasing its temperature after removing the source of heat? I watched an experiment and I'm not sure why this happened.
A metal rod was heated by a flame like this:
sensor
======== -> metal bar
flame

The metal sensor was on top touching the steel bar and a flame was heating the bar on the bottom side.
After heating up to more or less 140º Celsius and dilatation (metal expansion?) rate slowed, the flame was removed. 
The sensor that was reading the temperature by touching the metal bar (top) kept reading a temperature increase for 15+ sec. Why did this happen?
Was the heat being transferred to the cold contact region or something else? Does inertia have anything to do with this?
 A: Temperature doesn't have a mechanical-like inertia (in the sense of following Newton’s laws) or momentum; barring a chemical reaction such as combustion, you can't use a flame to heat up any material to a temperature greater than the adiabatic flame temperature, regardless of the heating rate. In other words, temperature never overshoots the temperature of the heating source.
One possibility, as you've noted, is that a cooler area of the bar/sensor continued to be heated by a hotter area of the bar even after the heat source was removed. In this case, the temperature would continue to increase—but it wouldn't exceed the flame temperature.
A: It probably has to do with the thermal diffusivity $\alpha = \kappa/(\rho c_p)$. It is equal to $\frac{\frac{\partial T}{\partial t}}{\nabla ^2 T}$ when convection and radiation effects are neglected. This latter formula gives some insights. The faster the temperature changes compared to its curvature (or the divergence of the gradient of T), the greater is the thermal diffusivity. So, from your description, all seems to point that the thermal diffusivity of the metal (steel?) is low enough to ensure that the cold side will still have an increase in temperature even about $15\ \mathrm{s}$ after the hot source was removed. 
Also, I do not agree with the claim that temperature does not have inertia. For example, the heat from the Sun will diffuse through the Earth ground in such a way that it is possible to dig a few meters below the ground surface and be able to spot summer times and winter times. But the amplitude of the temperature variations decays exponentially with depth. At large depths (something like above 20 m or so) only the average surface temperature is still distinguisible. By digging deep enough, it is possible to know the average surface temperature up to hundreds of years or even more (I remember several articles about that, I might try to provide references later). 
The above example is relevant, because it shows that even if the Sun was suddenly removed, the heat from past summers would still diffuse deeper and deeper into the ground, heating colder parts, well after the heat source (Sun) is removed. The same thing is probably happening from what you describe with the heated bar under a flame suddenly removed.
To summarize, you've noticed a peculiarity of conductive heat transfer. This is normal and would happen with any solid. The fact that the temperature increase lasted for about 15 s after the heat source was removed is due to both the geometry of the material and on its thermal properties, in particular its thermal diffusivity. A material with the same geometry than the steel rod but with a higher diffusivity such as copper would display a shorter such time, and conversely: a material with a lower diffusivity such as glass would display this temperature rise at their colder side for longer.
A: After the heat source is removed, the bar starts to cool down immediately. However, this only means the average temperature of the bar is decreasing. As long as there is a temperature difference, heat will continue to flow from the hotter to the colder regions.  
At some points the indicated temperature starts to decrease, this is the point where the material at the sensor looses more heat to the surrounding cooler material than it receives from the hotter areas.
