Can you put a cast iron teapot on a candle warmer?

Yes. Enamel-lined cast iron teapots are commonly sold with tealight warmers, which are the recommended way to keep them hot.

Can a cast iron teapot go on the stove?

Enamel-lined cast iron teapots (tetsukyusu) should not. Unlined tetsubin kettles are designed for direct heat. Check what you have before heating it.

These get mixed up constantly, and it matters for how you heat them.

Tetsubin is a traditional Japanese cast iron kettle for boiling water, usually unlined and made to go over heat.
Tetsukyusu is a cast iron teapot for brewing. It's usually lined with enamel. Most "cast iron teapots" sold online are this kind.

An enamel-lined teapot should not go directly on a stove burner, because the enamel can crack. A tealight warmer is the way to keep it hot.

Match the size. The warmer's ring should support the pot's base fully. Cast iron pots are heavy, so wobble matters more.
A sturdy, wide footprint so a full iron pot can't tip.
A matching style. Many teapot ranges sell a warmer in the same pattern, which guarantees the fit.

Rinse with hot water only. Skip the soap and the dishwasher.
Dry thoroughly, including the outside base, so it doesn't rust. Leave the lid off to air dry.
Never let it boil dry on a warmer, and don't plunge a hot pot into cold water.