Conic sections are a frequent target for dropping when attempting to make room for other topics in advanced algebra and precalculus courses. A common argument in favor of dropping them is that typical first-year calculus doesn't use conic sections at all. Do conic sections come up in typical intro-level undergraduate courses? In typical prelim grad-level courses? If so, where?
Conic sections are basic examples in algebraic geometry, since they are (the real forms of) curves of genus zero. As such, they are also basic examples in number theory, since it is easy to determine the rational points on a conic section, and this is a good warm-up for studying more complicated Diophantine equations. In fact, curves of genus zero are the only class of variety for which an algorithm provably exists to determine the rational points! Even for the next hardest case, elliptic curves, there are no algorithms which provably always work.
A great survey of these topics is Bjorn Poonen's Computing rational points on curves.
Edit: There is also Franz Lemmermeyer's Conics - a poor man's elliptic curves, which explains how certain conics can be thought of as "degenerate" elliptic curves.
If you're a physics sort of person, conic sections clearly come up when you study how Kepler figured out what the shapes of orbits are, and some of their synthetic properties give useful shortcuts to things like proving "equal area swept out in equal time" that need not involve calculus.
The other skills you typically learn while studying conic sections in analytic geometry - polar parametrization of curves, basic facts about various invariants related to triangles and conics, rotations and changing coordinate systems (so as to recognize the equation of a conic in general form as some sort of transformation of a standard done), are all extremely useful in physics. I'd say that plane analytic geometry was the single most useful math tool for me in solving physics problems until I got to fluid dynamics stuff (where that is replaced by complex analysis).
Relatedly, independent of their use in physics, I think they're a great way to show the connections between analytic and synthetic thinking in math, which will come up over and over again for people who go on to study math (coordinate-based versus intrinsic perspectives, respectively).
Conic sections should definitely be retained. If you don't cover conic sections, then what other examples can you cover?
Lines? Too simple. General curves? Insufficiently concrete.
Examples are very important for illustrating the general theory and techniques.
Also, in a multivariable calculus course, typical examples will involve quadric surfaces. Here conic sections will come into play, since hyperplane sections (or "level curves") of quadric surfaces are conic sections.
The study of parabolas (with axis parallel to y-axis) is useful when you have to solve 2nd degree inequations.
Apart from what Jamie Banks has said, conic sections are also a great way to introduce the application of matrices and determinants to examine the family of curves. It gives intuitive idea about what the eigenvalue and eigenvector are supposed to mean. Also this way of studying the curves will be useful when they study the multi-variable calculus.