Light from a normal point source travels omnidirectionally, so a given number of photons(1) traveling outward from a light source must illuminate a larger sphere the further you get from the source.

One way to intuit the law of inverse squares is to think about putting a postage stamp on a basketball, then putting the same postage stamp on a beach ball with double the radius. A fixed area on the surface of the sphere constitutes an ever decreasing percentage of the sphere.

Math Nerd Break

Because the formula for the surface area of the sphere is 4πr² the portion of the sphere covered by a fixed area n would be n/4πr². We can further simplify this by setting n to 4π and we get 1 * 1/r² (the inverse square of the radius). If we set n to 2π we get 1/2 * 1/r². This works no matter what size n is. And since everything else is constant, we can simplify it to for any given area, if you double r you get 1/r² the light.

This, of course, only works for omnidirectional point (or near point) sources of light.

This gets muddied up in lenses. (Let us assume for the moment we could design a lens with no reflection or refraction and perfect transmission.) Once the light hits the front element, it ceases to travel along its original trajectory. Inside the lens it is getting bent and redirected do it can be focused down to a point again.

Alan's excellent answer is in reference to the entire image area, but let us consider a point source that the law of inverse squares would apply to (that being the reflected light of of any given point on your subject).

This diagram shows 3 point sources of light going through a hypothetical perfect single lens system and focusing onto the image plane. Once the light from a given point hits the surface of the lens, it stops "spreading" and starts to converge, so the law of inverse squares no longer applies.

A more distant point will supply a smaller percentage of its light to the element, but it will also have more neighboring points focusing down onto the same sensel or grain, so there is very little effective difference. This is why you don't need to adjust your settings if the distance to your subject doubles (and your focal length does not).

As Alan (very correctly) points out, taking a relatively smaller total image area does reduce the amount of light available, hence telephoto lenses having larger front elements to capture a larger area of the light cast off the subject. This is also why "fixed aperture"(2) telephoto lenses are built to relatively adjust(3) how much light they are blocking with the aperture so they can maintain a constant exposure value across all the points in the image. "Normal" zoom lenses do not perform the complex ballet of element movement to achieve this, but it is the trade off of weight, cost, and complexity that is the cause of the lost of light as you zoom on these lenses, not the law of inverse squares due to the distance traveled within the lens.

So I think it is safe to say that no, the law of inverse squares is not at play inside a lens in any appreciable way. All the math still checks out if you think of it from the perspective of the imagable area of point sources passing through the lens (the behavior of the image is a cumulative reflection "inside" the lens of the actual individual point sources outside the lens). But no, the loss inside the lens is not from the distance the light travels.

tl;dr

Once a point source of light hits the front element, you can effectively ignore the law of inverse squares because the light is no longer diverging from the source, but rather converging onto the image plane.

1. or if you prefer, the intensity of the waves
2. most (if not all) telephoto lenses maintain a fixed physical aperture as they zoom, so this is a bit of a misnomer, "fixed f-number" would be more accurate
3. by manipulating the shape of the image cone passing through the physically fixed aperture opening, not by adjusting the size of the aperture opening