According to the inverse square law, is the intensity at the source always infinity?

This is just an artefact from assuming your heat source is infinitesimal in size. Also, more generally, physics tends to break down in the limit $r\to 0$.

So, in many cases, you have the luxury of doing 'something else' in the vicinity of $r=0$. For example, it is common in various simulation settings (n-body, free energy perturbation, etc) to use so-called soft-core potentials. These are potentials that are modified so that the singularities disappear.

For a $1/r^n$ potential, it's common to replace it with something of the form $$ \frac{1}{(1+r^{ns})^{1/s}} $$ Here is a plot for $n=2,s=3$.

The inverse square law applies to point sources. In nature there are no point sources, so strictly speaking it never applies.

In nature we have extended sources. If you are far away from the source, then it "looks like" a point source, so the inverse square law is a good approximation. Close to the source, the inverse square law doesn't apply unless the source happens to be perfectly spherical. Even in that case the law breaks down once you cross the surface and are inside the source.

Update

@AnonymousCoward (below) reminds me that electrons do appear to be point sources. I point this out for completeness. This fact doesn't change things for your question, though.

Yes. An inverse square law is a law stating that some physical quantity is proportional to $1/r^2$, i.e. $X = Y/r^2$ is the general formula.

And so, the limit as distance [$r$] approaches 0 is "equal to infinity".

That is to say, the strength of a force that obeys the inverse-square law (like the electromagnetic force) approaches infinity as the distance to the source of the force tends to 0.

NOTE: The distance term in the inverse square law, usually denoted $\mathbf{r}$, represents the distance from the source, which is why right AT the source, $r=0$, so the magnitude of the force would be something divided by 0, which would be 'infinity'.

So two electrically charged particles with the same charge (i.e. both positive or both negative), for example, cannot 'touch' as it were (unless other forces were brought into play), because the electromagnetic force is an inverse-square law force: $$E = \frac{kQq}{r^2}$$