I don't mean to be pedantic, but a headphone amp amplifies the electric current. Up to a certain level and depending on your source, the drivers of your headphones, and the amp itself, this increase in current will most likely yield appreciable and desirable modulation of the sound. In a nutshell, kiteki gave a perfectly acceptable and very newcomer friendly response, but I just wanted to hop in for the sake of clarity.
The reason that many people here at head-fi find headphone amplifiers desirable or necessary is that some headphones (and this is more common with higher end headphones) require more electrical current than most portable devices or even some desktop sources are capable of supplying. This additional current that is introduced to the signal by the headphone amplifier allows the source to more effectively control the transducer. Resulting in more accurate sound.
The scientific evidence for burn-in is, to say the least, not compelling. But many people are believers and there are logical arguments in favor of certain aspects of what people label "burn-in". Burn-in is often a hotly contended subject, and many different listeners will define the phenomenon in different ways. First there is the material definition. Some listeners insist that running current through hardware or even cables will change the sound of that hardware or those cables over time. Others insist that it's not the cables or hardware, but the material that the drivers are made of, flexing and being put under duress, that make appreciable changes to sound. Others will insist that this is all BS. There is also the psychological definition where burn-in just refers to the way our brains gradually become accustomed to different sound signatures. It's not the amp, the headphones, or the cables changing, but rather the ever evolving way that we perceive them. Some people incorporate both definitions and will use the term 'burn-in' to refer to both material and psychological changes (which is sensible, as if one were to believe that both were real then the two would also be inextricable).
Once you take into account that since you can't listen contemporaneously to the same exact amp, source, cable, and/or pair of headphones as they existed at different points in time, comparisons must mainly be made from memory, which is in many ways truly immeasurable and quite fallible. There are simply too many variables in human perception (the reason we have conflicting witness testimonies in legal matters, especially surrounding traumatic events) and not enough instruments of measurement to say for certain whether burn-in exists empirically or not.
With this understanding of burn in we can now interpret burn-in the way we would a placebo or an athlete's pre-game ritual superstitions. Hell, even I sometimes speculate on whether it's my brain, ears, or listening equipment that has changed sometimes when I notice different sensations from the same hardware and recordings.
If leaving your new cans in a box running pink noise at your normal listening volume for 200 hours straight makes you appreciate them more and ascribe more value to what was probably an expensive head-fi purchase, then more power to you. I will not take that away from you. But there are many sanctimonious head-fiers out there that don't take quite the same live and let live approach that I do, and some of them will assure you that burn-in is definitely real; others will call you a fool. Whether you believe in it and by extension whether you join one of the aforementioned camps about its existence is up to you. Try the pink noise thing. Maybe you'll notice a difference. Maybe the difference was just your brain getting used to not listening to them while they were burning in. If you don't notice a difference, maybe you've disproved the existence of burn-in. Maybe you've just proven that your ears aren't or your equipment isn't suited to the perception of a difference. The greater point I hoped to achieve with these illustrations is that there are plenty of arguments for and against the existence of burn in that will carry varying levels of credulity with different sects of head-fi, for whom the same events and perceptions could be interpreted differently based on belief. If you don't want to have to slog through even more long-winded versions of what I've just laid out, then it's best just not to mention 'burn-in'.