Does a warning light on the dashboard fail an MOT?

It depends which light. A handful of specific warning lights are actually checked as part of the MOT and can fail the test if they're on, or not working at all - the ABS light and the engine management light are two of the main ones.

Others, like a general airbag warning light, aren't automatically a fail in every case - a tester is generally looking for whether the light is showing a genuine problem, rather than failing a car simply because the light is present.

In practice, if a warning light comes on and stays on, it usually means the underlying system has logged a fault, and it's that fault, not the light itself, that determines whether the car fails. It's worth getting any warning light checked regardless of whether it happens to be a specific MOT test item, since the underlying issue doesn't go away just because it doesn't fail the test.

Source: gov.uk: car parts checked at an MOT