Question: Do Ivy League schools reconsider a student's application once he or she has been denied in regular decision? Are there any specific situations in which a student can ask for reconsideration? How can one approach this?
Colleges, especially hyper-competitive schools like the Ivies, do not reverse denials. If they did, they would be deluged with appeals every year from the many thousands of terrific and qualified students they reject. It's kind of like the contest-rules phrase, "Decisions of our judges are final."
If you feel that there is a reason that you should be reconsidered nonetheless, you can certainly pursue this avenue, although your situation would have to be truly exceptional. What do I mean by "exceptional"? For example, are you convinced that your guidance counselor submitted the wrong transcript--confusing your grades with those of a less able student whose name is the same as yours? Did you withhold highly uncommon details about yourself (e.g., you are a quadriplegic or totally deaf or blind) because you initially didn't want this impacting your admission outcomes? Have you won Teen Jeopardy, published a best-selling novel, or achieved something similarly extraordinary since your application was submitted? Such circumstances might be considered extenuating enough for you to launch an admission-denial appeal, but--even then--you'll have an uphill battle.
Some institutions--particularly the larger public universities, whose verdicts heavily weigh numerical statistics rather than more subjective factors--actually offer a formal appeals process. Certainly, when seeking a decision-reversal at such a school, that's the place to start. However, with the Ivies, there is no formal route to an appeal, and I caution you not to spend your time on one unless you feel that your position is truly unique.
If you do, you can begin with an appeal letter that clearly explains why your case is worthy of further consideration. Typically, when we at College Confidential assist students who are looking to move from a waiting list into the "admit" pile, we suggest that they provide the college with new data about how well they have performed since their applications were submitted (assuming, of course, that this performance has been markedly better). However, this sort of update is not significant enough to prompt a denial-reversal. As I said above, the letter will need to reveal truly exceptional information.
While you are surely disappointed that you did not receive the news you wanted from your favorite college or colleges, please keep in mind that there are many roads to personal and professional fulfillment, and that a "rejection" says nothing about what kind of person you are nor about the successes that lie ahead for you. I urge you to consider the other options that await you and allow yourself to get excited about them now.