For the purposes of this discussion, we will rule out talking about Yixing clay tea pots since they are specifically made to take on the flavor of the teas brewed in them.
We are talking about the other 99% of pots that most people use - ceramic, glass, iron, steel or porcelain.
What material is best for which teas?
Not a huge amount of pots are made from stainless steel, mainly because they are costly. Some people say that certain drinks taste better out of a bottle than a can. The craft brew industry uses cans, not to mention the entire beer industry brews in stainless steel. Almost all meals prepared in restaurants are made using stainless steel vessels. So the taste difference may be more of a perception issue than reality.
Porcelain and ceramic are both made from clay. The difference is that porcelain is more refined and purified, making it harder and offering more design flexibility. That is why fancier cups and pots are made of porcelain, allowing them to make thinner and more elaborate designs.
We all know glass, and the teapots made from glass are generally on the thin side. Aside from a level of fragility, they are flavor neutral.
Iron pots are coated with enamel, and thus the metal never comes in contact with the tea.
From a pure taste perspective, none of the aforementioned materials will make any difference with regards to the taste of the tea.
But what about heat? There are some that recommend certain pots based on their thermal properties. For example, glass pots are generally thin so they do not insulate and therefore lose heat faster. Some say green tea would be better suited in a glass pot since it is consumed at lower temperatures versus black tea.
Worried about insulation?
Iron has a higher heat transfer rate than ceramic. This is true, although you still can't pick up either at the bottom with your bare hands. One writer mentions " Due to their composition these pots transfer heat easily (they very quickly lose the temperature of the tea liquor)". This is true only if you pour hot water into a cold iron pot. Think of a cast iron radiator. It holds heat for hours right? It takes a while to heat up, but once all that metal is heated - it stays warm. So simply filling up an iron pot ahead of time with hot water will 'prime' the pot. Once warmed up, your tea will stay warm for a longer time.
Heavy ceramic pots can also be pre-warmed, especially during the winter. We tested an insulated ceramic pot with a built in tea cozy. The actual difference was minimal.
The only way to keep tea warm for that long a time is to use a tea candle under the pot or use an insulated stainless steel tumbler. Tea candles pose a problem as well, after a period of time the tea will cook and change flavor. This depends on the tea of course, with black tea being more resilient under the flame.
Overall, the pot you NEED to use is up to you!
In practical terms, regardless if you drink green or black tea, and assuming you are not brewing in an ice cold pot, your tea will taste exactly the same when brewed in any of these pots mentioned. Even if you switch tea types, assuming you wash the pots you should not taste remnants of previous brews.
Ultimately, there is no reason to over think this. Do what feels natural. Would you drink a nice vintage of wine in a thick water glass? No. The same applies to tea. Brewing a pot of Gyokuro in a thick and stout ceramic pot will taste exactly the same as glass, but some teas have great color that can be shown off in glass.
For new tea drinkers, a general purpose ceramic pot is just fine. And as your tastes expand, you'll find yourself compelled and drawn to certain pots for certain types of teas. Use the pot that feels naturally good for the tea you are drinking.