Bending a mattress can cause damage and may affect the integrity and overall comfort of the mattress. It's not recommended to bend a mattress as it can damage the internal springs, foam layers, and other components that make up the mattress. (Any mattress that is designed to work on an adjustable bed, like many of ours, can be safely bent to at least 45 degrees) Additionally, bending a mattress can also cause the materials to lose their shape and support over time (if you only bend a bendable mattress once it's never going to be an issue over the long term, ever).
If you need to get a mattress around tight corners, it's best to disassemble the bed frame or remove the door from the room to ensure that the mattress can fit through without bending. (The bed frame has nothing to with a mattresses ability to go around tight corners. The best mattress type for this purpose is all foam but not super hard)