You can put air dry clay in an oven and it will not catch fire, which is why the advice circulates. What it will not do is become ceramic, and what it very often does is crack. The material hardens by losing water, not by a chemical change at temperature, so an oven adds risk without adding the thing people are hoping for.
What an oven does to this material
Domestic oven temperatures drive water out of the outer few millimetres far faster than water can migrate from the centre. The surface shrinks against a full-size interior and the piece cracks, usually through the thickest part and usually after it has cooled, which is why people report that it seemed fine at the time. Nothing about the binder is improved by heat, so the whole exchange is risk for no gain.
What people are actually trying to achieve
Two things, and neither needs an oven. If the goal is speed, moving air and a thinner or hollowed form get you there without the cracking. If the goal is a hard, water-resistant object, no oven temperature available in a kitchen will vitrify a clay body: that needs roughly 1100 degrees and upwards, which is a kiln. Between those two goals sits the whole question, and an oven answers neither.
The one case where low heat is defensible
A gently warm, switched-off oven used as a draught-free warm cupboard, with the door ajar and the piece well away from any element, is a slow drying environment rather than a bake. Even then the gain over a room with a fan is small. If you use it, treat it as storage rather than as a process, and never turn the oven on with the work inside.
If you want fired, buy fired
The pottery suppliers priced on this site sell stoneware and earthenware bodies from roughly a pound a kilogram excluding VAT, and many studios will fire other people's work for a fee. That route costs less than most people assume and produces an object that holds water, survives a dishwasher and can carry a glaze. Air dry clay is the right material for a great many jobs, and this is not one of them.
Questions people ask about air dry clay
Can you bake air dry clay?
You can put it in an oven, but it will not become ceramic and it is likely to crack. The material hardens by drying, not by heat.
Can you put air dry clay in the oven to speed up drying?
It speeds up surface drying and leaves the core wet, which is exactly what causes cracks. Airflow is the safe way to accelerate drying.
What temperature would actually fire it?
Around 1100 degrees Celsius and upwards, which is a kiln rather than an oven, and most air dry bodies are not formulated to be fired at all.