Are there any halachos which require a tallis to be of a certain color? I've seen talleisim of many different colors, but are there any actual rules with regard to the color as opposed to the material?
Tosfos writes that there is no obligation to do so, and the color of the strings does not have to be the same as that of the Tallis. The Halacha is like this opinion. Moreover, the custom in the Ashkenazi world is to make the strings white even when the garment is another color. The reason why Ashkenazim make the strings white is because it says in the Gemara that Hashem dressed himself up in a Tallis, and it says in a verse in Daniel that "His garment was as white as snow".
Therefore, to keep to all opinions, we make both the strings and garment white.
The Pri Mgadim the lines on bottom of the Tallis (which is there to remember the Tcheiles which we don't have) is not an issue because we go after the color of the main garment.
The Eishel Avrohom from Butchatch writes that it is fine if the garment aged and turned yellow.
R' Shmuel wrote an excellent, well-researched answer, but I think he may have misunderstood R' Naftuli's question. Most modern talleisim are in fact white, with colored striping. Sometimes the striping will be fairly conservative (e.g. blue) and sometimes it will have a more modern look (e.g. purple and gold striping or green or orange or rainbow striping). So the tzitzis and the beged are actually both white.
This is reflected in the Mishnah Berurah (O.C. 9, 5, 16), based on the Pri Megadim, where it says בתר עיקר הבגד אזלינן (we go according to the color of the majority of the garment). He writes this to explain the fact that the edge of the beged is a techeiles color in our talleisim.
This remark has always been mystery to me. I have read that the striping on the traditional tallis gadol morphed from blue to black at some point. I can't even figure out for sure whether he is referring to a tallis katan or a tallis gadol.