Never put a teapot into the dishwasher or immerse in soapy water which might leave a soapy residue affecting the taste of your tea. Pour out the tea, rinse the pot with clean water, and turn it upside-down to drain. Dry the outside only.
If the inside of your china teapot is stained, just toss a denture-cleaning tablet into it and pour in warm water to cover the stain. Let it sit for an hour or so, then rinse with cool water. Use a small nipple brush (find these in the newborn baby section) to clean out the spout.