Light's inverse square law: Does it require a minimum distance from the source? Does the inverse square law begin to take effect the moment light leaves its source? For example, does light's intensity decrease, i.e. does the area in which the photons might land increase, at a few millimeters from the source?
I happened to come across an article about emergency lights and photometry from a few decades ago that appears to answer in the negative:
"The minimum test distance in photometry of these sources is called the 'minimum inverse-square distance.' The illumination from the light source, measured at distances greater than this minimum, obeys the inverse-square law which is a necessary criterion for the determination of luminous intensity. [...] The minimum inverse-square distance is determined by the type and size of the light source, lens, reflector, etc., and must be considered individually for each unit. If this distance is more than 100 meters (approximately 328 feet), a ranger larger than 100 meters must be used."
Source: Howett, et al. 1978. "Emergency vehicle warning lights: state of the art." USDC. NBS Special Publication 480-16.
 A: The inverse square law applies to point sources. A real emergency light is not a point source, and therefore the law appears to not apply at close distances, because any real point is at a varying distance from different parts of the emergency light.
A: The inverse square law says that the intensity of incident light falls off in proportion to the inverse of the square of the distance from the light source.
The important word here is "the distance" — the inverse square law implicitly assumes that all parts of the light source are at the same distance from the measurement point, or at least approximately so.  For real-world light sources, which are not infinitesimally small points, this approximation must necessarily fail when you get close enough to the source — you can pick a measurement point arbitrarily close to some part of the source, but you can't get it arbitrarily close to all parts of the source at the same time.

So, how close is too close, then?  For that, we can come up with all kinds of rules of thumb (like, say, "no closer than $x$ times the maximum diameter of the source", for some value $x$), but if you want a precise, quantitative answer, we're going to have to do some math.
For simplicity, let's consider the (in some sense worst) case where the extended light source consists of two identical, very small pointlike light sources spaced a distance $2d$ apart.  (We assume that the diameter of the individual point sources is very small compared to the distance $d$, so that it can be safely neglected.)  We'll take the midpoint between the two point sources (i.e. at distance $d$ from each of them) as the nominal center of the extended light source, and place our measuring device at a distance $r > d$ from it.
Let's first look at the case where the two point sources and the measuring device are all collinear (or just very slightly staggered, so that the two point sources won't eclipse each other).  Then one of the point sources will actually be at the distance $r_1 = r - d$ and the other one at the distance $r_2 = r + d$ from the measurement point.  Thus (since each individual point source is assumed to be negligibly small, and so follows the inverse square law very closely) the combined intensity of the light from the two sources will be proportional to:
$$\frac12 \left( \frac1{r_1^2} + \frac1{r_2^2} \right)
= \frac12 \left( \frac1{(r - d)^2} + \frac1{(r + d)^2} \right)
= \frac1{r^2 - d^2}
\\\approx \frac1{r^2} \left(1 + \left(\tfrac{d}{r}\right)^2 + \left(\tfrac{d}{r}\right)^4 + \dots \right)
$$
where the dots denote higher-order terms ($O(\frac{d^6}{r^6})$ and higher).
We can also look at the opposite case, where the line between the point sources is perpendicular to the line from its midpoint to the measurement point, so that by Pythagoras' law, $r_1 = r_2 = \sqrt{r^2 + d^2}$.  Then the actual light intensity at distance $r$ from the midpoint is:
$$\frac12 \left( \frac1{r_1^2} + \frac1{r_2^2} \right)
= \frac1{r^2 + d^2}
\approx \frac1{r^2} \left(1 - \left(\tfrac{d}{r}\right)^2 + \left(\tfrac{d}{r}\right)^4 - \dots \right).
$$
In both cases, the relative error in the $\frac1{r^2}$ approximation is approximately proportional to the square of $\frac dr$ (and the absolute error is thus inversely proportional to the fourth power of $r$), although the sign of the leading error term is different.
Other configurations of point sources (with the same maximum diameter $2d$) will generally fall somewhere in between these two extreme cases.  Thus, when the distance $r$ to the light source is, say, 10 times the half-diameter $d$ of the source, we can pretty confidently say that the relative error in the light intensity calculated using the simple inverse square approximation, compared to the true intensity obtained by integrating over the full extended light source, is at most $\left(\frac{1}{10}\right)^2 = \frac{1}{100} = 1\%$.
A: The quote from the reference says it all: (I added caps) "The minimum test distance IN PHOTOMETRY of these sources is called the 'minimum inverse-square distance.'"
The minimum distance is therefore a photometry issue, in other words, a measurement problem.
The essence of the measurement problem is how far away you have to be before you can approximate the light source as a point source. 
That is the minimum distance.
A: Cort and Ilmari have given good answers about the practical issue: the inverse square law is for point sources, and so a non-point source (like an emergency light) will only appear to have the same properties at some minimum distance that depends on the geometry of the real source.
However, it seems nobody has mentioned a different "minimum distance" that applies to even point sources (such as the electromagnetic field generated by a single electron). It turns out that in quantum electrodynamics (QED), the electromagnetic gauge coupling (which determines the strength of electromagnetic forces) is only approximately constant. At very high energies (corresponding to very small distance scales), the coupling strength increases, so that photons at these scales would appear to not obey the inverse square law, but would instead appear to lose their "brightness" even faster. This is of course not at all relevant on the scale of things like emergency lights, but rather on scales even smaller than the proton.
A: As many have said, the inverse square law applies to point-sources.  These are idealized light sources which are sufficiently small compared to the rest of the geometry that their size is of no importance.  If a light source is larger, it is typically modeled as a collection of idealized light sources, potentially using integration.  The exact definition of "sufficiently small" varies with application.  The definition of a "point source" for astronomy is quite different from the definition of "point source" for a LCD projector.
There is actually a limit to this process.  The inverse square law is only valid in its normal form if you are working on scales where light can be modeled purely as a wave.  As you get very small, on the microscopic scales, those assumptions break down.  You instead have to think about the statistical expectation of photons, which follows the statistical analogue of the inverse square law.  Even smaller, and you start to enter the world of quantum mechanics, where you have to account for the actual waveforms of the objects under study.
Ignoring these corner cases, nearly all cases you find will have "sufficiently small" defined by macroscopic factors, like the sizes and locations of lenses.  Its rare to find oneself in the world where the microscopic factors matter.
A: The inverse square law applies to point sources.  For extended sources becomes accurate at distances that are large compared to the size of the source.  At large distances the source looks like a point.   What "large" means depend on the application.  In the case of light fixtures, the Illuminating Engineering Society and other organizations have made judgments about what is large and what is not based on the use case.  Is it room lighting?  Is it illumination of products in a grocery store?  Etc.  There are published advice and tables to guide the lighting designer.
