Not at all. First of all, about 0.014 of the sun's mass is composed of Fe. If you do the math, you will realize that is over 4660 Earth masses of Fe in the sun. This is because the sun is a relatively high metallicity star (population I) and was likely formed in a zone entrenched with supernovae products. If you just look at the constitutents of the inner planets, asteroids, and meterorites, this is in line with what would be expected.
If a "large" mass of Fe (comets don't contain much metal but we could replace your proposed comet with an asteroid of similar dimension) was to fall into the sun, it would just completely vaporize. The sun's temperature at the surface is ~5778 Kelvin which is much higher than the boiling point of Fe, Ni, or any alloy of the Fe-Ni. The atoms would disperse and would become lost in the sea of H and He nuclei, and wouldn't conceivably come anywhere near the core in one piece. Literally nothing about the sun would change in this scenario. Metal-rich objects (small ones) likely fall into the sun all the time and aren't detected.
Now for some comments on fusion. The sun first of all won't ever fuse Fe in it's core as it simply isn't massive enough to do so. The temperatures required to fuse increasingly heavy nuclei have to increase enormously and this is aided by the core contracting and heating from the enormous pressures and masses pressing down from above. The sun simply doesn't have what it takes.
Also, Fe doesn't fuse not because the nuclei are too heavy. Fe in a massive core will indeed fuse into Ni and could go conceivably further. The problem is that these reactions are endothermic meaning they will absorb energy. At this point too, the temperature is so high that photons are able to split Fe and Ni nuclei apart back into lighter nuclei and individual nucleons. All of these processes will take energy out of the core. The less energy in the core, the more it will contract, the hotter it will get, and energy is lost faster until the core succumbs and collapses. Note that this is not an instinteneous event happening just from the presence of Fe. As the core collapses further, protons and electrons find it energetically favourable to form neutrons (neutronization), which takes more energy.
I'm assuming you watched a program where they said that "iron is a star-killer" or some other dramatic nonsense like that. Remember that shows on television are made with entertainment value in mind and will simplify or simply misinterpret things for a general audience. Always watch with skepticism.