Why Yixing?

First, why is it typically limited to Oolongs and Pu’er?

One of the benefits of an yixing clay teapot (over, say, a gaiwan) is the fact that they hold heat incredibly well. When you pour boiling water over and into your yixing, the clay holds the heat and, in turn, is also very close to that high temperature. It also helps to keep the water from cooling too quickly during your steeps. The higher quality the clay and the better the fit between lid and body (craftsmanship), the better this will work.
You probably wouldn’t want this kind of environment for green teas or tea that otherwise prefers a cooler temperature. That is the main practical reason yixing is not typically used for greens. Because of the lower temperature required, I typically make greens in glasses (where the cool water can cool down even more quickly) or, if I’m using a big Western pot, I steep with the lid off (again, to help with the temp).

There is no real reason not to use a teapot for Indian black teas (for example) other than tradition. Some folks might fear that the texture imparted by the pot into the tea might not mix well with an Indian tea, but really? You never know until you try. There is also the slight issue of Indian blacks generally being very small and including fannings, etc, and some might worry that these could clog in the spout or get into your tea cup. This can be mitigated by using a metal strainer over your pitcher, and by thoroughly rinsing your pot out after every use with very hot water and no soap.
If you ever did get small tea particles stuck in the spout and couldn’t rinse them out, there would be a slight danger of mold. :( Just something to watch out for.
Another thing you’ll want to watch for is the pour-rate of your teapot. If you’re using it for an Indian black, the tea might be a little more sensitive about turning bitter or dry. Check to make sure your pot pours out relatively quickly with respect to its volume (these guys will usually have a bigger spout connection to the pot, or a wider hole) so that you’re not pouring so long that the back end of your pour ends up being over-steeped.

So no- no real reason to use a pot for only oolongs and pu’er apart from:
- Water Temperature
- Leaf size/shape fitting with the size/shape of the teapot (ie: a Da Hong Pao would need a pot with a large lid… a Keemen pot wouldn’t, but you’d need to watch the spout for itty bitty leaves).
- Tradition

Other than those, any tea you can make in a gaiwan, you can make in an yixing. Any tea that might benefit from gong-fu style short steeps with more tea in less water could instead use an yixing pot.

Re: Not having to use tea in an yixing over time
Yes, it would take years and years of heavy use to get to the point where you wouldn’t need to put tea into the pot.
However, this is kind of an exaggeration.. it’s not really why people use yixing (well, I certainly don’t use them so that I don’t have to use tea leaves at some point in the future).

People use yixing for both practical and aesthetic reasons. Let’s get the practical out of the way first. As you’ve heard, yixing does give the flavor of whatever tea you’ve been making in the pot back to all future teas you make in the pot. This doesn’t exactly amplify the flavors of your tea, nor does it truly replace tea leaves. Instead, it deepens the flavors you’d get out of the tea and makes them more complex. In the pots I’ve been using less (total actual use time is about 6 months even though I’ve had them over a year), the main difference is in the texture imparted to the tea. The mouthfeeling becomes very complex and interesting; it’s hard to explain.. the effect really depends on the pot and the tea.
For example, I’ve been wanting to dedicate a pot to Dan Cong oolong. At the moment, I have two pots each for Da Hong Pao/Roasted Oolong and Anxi Tieguanyin/Green Oolong. I want to transition one of these pots over to Dan Cong (I still don’t know which!). The green oolong pot brings out the greener, more apple-y and juicy characteristics of the Dan Cong. The texture is also smoother and generally more on the sweet end. Compared to doing the tea in a Gaiwan, the steepings are more like the first half of steepings, but with less bouncy grassiness.
In the Da Hong Pao pot, the tea suddenly has more of a stoney texture.. hard to explain, but it has a sweetness I associate with clear stream water.. it’s like one of those perfectly smooth rocks has been placed in the pot from the bottom of that stream. The flavors also now lean more towards what the tea is like in the latter half of a session in a gaiwan. More woodsy, more of a mellow stick flavor… something that reminds me of s’mores.
It’s clearly the same tea each time, but the character has certainly changed in a definite, noticeable, and unique way. The main changes are in the texture and mouthfeeling, and they are related to the kinds of tea I’ve been making in the pot previously.

In pots I’ve been using a lot (like my sheng and sho pu’er pots), I’ve gotten to the point where no matter what tea I put into the pot, it tastes REALLY expensive.. very fine and extremely complex. This is because I usually only make my nicest pu’er in these pots, so those flavors are actually making guest appearances in teas I make in the pot now. It’s not like I’m making the two teas at the same time- it’s not that strong. But it’s as if the tea has taken on some of the characteristics of the other teas by association. So now any sho pu’er made in my really nice sho pot will have some sparkling, sweet old-library and magical well-water taste and texture to them. It won’t be the main taste, but it’ll be there, adding delicious layers of experience onto whatever the tea is naturally giving me.

Here’s one more practical reason I like yixing better than gaiwans. I have really tiny hands. Most gaiwans are just too big for me to use, so it’s hard for me to make tea with them with the ease I want. My gaiwans, even the teensy one I do have, also transfer heat very well to my fingers. Thus, burns. :( I have not yet burned all the feeling off in my fingers, so an errant slip the splashes water on my fingers makes for an unpleasant steep for me and any guests (woop! woop! woop! Oh no, I just spilled boiling tea all over the board and on your face. Sorry?). Because yixing pots have handles on them, these handles are easier for me to.. well.. handle.

But that’s just practically why I use yixing. I also use yixing pots for the very impractical reason of aesthetics.
I make tea because it’s an experience of the senses. Tea as an activity and a social experience, rather than tea as a thirst-quenching beverage. When I make tea gong-fu style, I am making tea for someone else, even if that someone is only a (hopefully!) better version of myself.
I prefer to use yixing pots when I make gong-fu style tea because they make the whole activity so much more special. They are beautiful. They require more of my attention to use, which helps me in turn to focus on the flavors of the tea I’m steeping in those pots. I have to take care of the pots by "yang"ing them, which makes them even more beautiful over time.
The yixing represent an investment on my part in beauty of all the senses. Because I have these pots, I am much more likely to want to make nice tea in a nice way, if only because I feel bad when I haven’t used them for awhile.
Basically, because the pots are objects of beauty, made with care and used with care, they impart that sense of beauty onto the tea I am drinking, so no matter what’s coming out of my pot, I’m going to enjoy it.

I hope you enjoy your pots, and that they make you good tasting tea! If you appreciate the pots for who they are, then enjoyment is practically guaranteed.

PS:
Apologies for the length. I do ramble on. Here are some other good threads with questions and info about yixing pots. I’ve got some lengthy opinions in there, but there are also great people who disagree with me and who outline their perspective well. It would be fun or you to hear all of those sides of this ongoing conversation.
http://steepster.com/discuss/2042-how-much-to-spend-on-yixing
http://steepster.com/discuss/1981-for-those-into-chinese-teapots

I have a yixng used for green tea only, because it’s too big for oolong or puerh. Yixing originated from green tea drinking region and was first used for green tea and red (black) tea. I think it’s ok to use yixing for any unflavored tea – even flavored tea, if someone wishes.

Spoonvonstup gave very good explanation on the reason to use yixing. Similarly, for me, aesthetic reason is the most important. Yixing, even the inexpensive (but authentic) ones, get more charming when more used. For most of my personal items such as shoes and bags, I tend to like older ones much better than new ones, because the older ones feel more “mine” :-D But many other things wear out with time being. Yixing don’t wear out, only gets prettier. Well they are fragile and I’ve broken quite a few. But that’s life ;-)