@Willis has a great answer which I don't entirely agree with. However, I do not believe there is a right answer so am not saying leaving boots on is wrong - its very situation dependent. Generally I would advocate removing boots unless there is good reason not to.
Leaving feet in boots means little movement, loss of blood flow therefore loss of heat. In such a situation of a force overnight bivy, frostbite is very likely unless you can maintain blood flow to the extremities - leaving boot on all night will almost certainly mean frostbite by morning. You need to get heat to the feet.
The only way to increase blood flow to cool extremities is movement - massage or exercise. With boots on, in a forced bivy, movement of feet and toes is extremely limited. With boots on, there is not way to add heat.
By removing boots, you can massage feet and use hands to get cold muscles moving and increase blood flow- you can also stick the feet under armpits etc (your mates, not yours) to obtain heat. Clearly this is not an option if solo, leaving blood flow as the only way to get the heat where its needed.
Unless you are extremely disciplined about looking after your feet, once you stop feeling them, you will not be bothered removing your boots, and sacrificing hand warmth in the process, to warm them up.