This question is more interesting than I thought at first. I like it. There are several different parts to an answer to this question; I'll just contribute a couple that have something in common: our bodies (and everything else, it has nothing to do with bodies) also emit photons about as fast as they absorb them.
On the macroscopic/thermal scale, we have black-body radiation. Via black-body radiation, all matter emits a continuous spectrum of radiation. The distribution of this spectrum depends primarily on the temperature of the object. This is why objects placed in a fire appear to glow red, whether they be wood or metal or rocks. Our bodies also emit radiation this way, but at our temperature, this spectrum is in the infrared range, so it isn't visible (to humans—snakes can see body heat). Since everything absorbs and emits photons this way, there's an equilibrium where we receive as much thermal energy as we lose, although that's only in an environment where everything is at equilibrium. 184
Hot things like the sun and incandescent lights can throw this off, which is why it feels hot to go outside... or to be under a heat lamp. Anyway, don't worry about filling up on too many photons, they leave you just as fast.
On the microscopic scale, we have the hard-to-spell phenomenon of fluorescence. When a high-energy photon is absorbed by an atom, some of its energy can be reemitted as a lower-energy photon. Of course, this doesn't happen every time, and I don't know if it happens much in our bodies. It depends on the material's properties. That's where we get fluorescent lights and pigments and laundry detergents—detergent manufacturers actually include fluorescent pigments in their products so that clothes emit more visible light than they physically should by absorbing UV light and reemitting it in the visible range. Anyway, while I'm not sure if that principle in particular saves you from atomization, it's worth remembering that not every photon that hits you stays there.
So in conclusion, even though the energy that light brings to our bodies (and the earth) is substantial (imagine if there were no sun—radiation is important!), we're not going to fill up on photons to bursting. We're at equilibrium. 104