The detergent line
Glass, porcelain, fully glazed ceramic, stainless steel and intact enamel are non-porous. Detergent sits on the surface, rinses off and is gone. Unglazed clay, bamboo, wood, unfinished stone and crazed glaze are not. They take detergent into the surface and release it slowly for a long time afterwards, which is detectable in the cup and is the commonest way a good pot is spoiled. If you are unsure which side of the line an object is on, look at the foot ring and the interior separately — partially glazed pieces are where people get caught.
What a tannin stain actually is
The brown film inside a teapot is polyphenol material deposited from the liquor, sometimes with mineral content from hard water bound into it. On a glazed or glass surface it is cosmetic and comes off with bicarbonate of soda in hot water, left to stand and then wiped. It matters in exactly one situation: when you are judging liquor colour, which is why professional tasting sets are kept white. On unglazed clay the same film is part of what people call patina, and removing it is removing the seasoning.
What never to do
Do not scour a glaze, which scratches it and makes it stain faster afterwards. Do not bleach crackle ware, which lifts the colour out of the cracks unevenly. Do not put bamboo, lacquer, wood or unglazed clay in a dishwasher. Do not use metal polish anywhere the tea touches. Do not push wire through a ceramic filter, which chips it and enlarges the holes unevenly. And do not scrub the interior of an unlined iron kettle, which removes the layer that keeps it stable.
Drying is the part people skip
More teaware is lost to being put away damp than to anything else. Unglazed pots grow mould if closed while wet, so they dry with the lid off. Unglazed foot rings hold water and mark a stack. Bamboo splits and mildews. Cane and lacquer are damaged by standing water. Cloth filters sour. Vacuum flasks sealed damp develop a smell that transfers to everything afterwards. In every case the fix is the same and costs nothing: dry it properly and store it open.
Smells, and what to do about them
A vessel that smells of the last tea has taken something into a surface. On glass and porcelain, bicarbonate of soda and hot water usually clears it. On plastic and silicone, often nothing does, and the seal is the part to replace. On unglazed clay, a long boil in clean water and then several weeks resting dry will reduce it and rarely removes it, which is why the one-pot-one-tea convention is worth keeping rather than testing.