To throw out a bed in Japan you have to arrange for a bulky waste pickup also known as a sodai gomi through the local city office.
The process of arranging for a bulky waste pickup to dispose of a bed or mattress or other large waste involves contacting the city in Japan's oversized garbage reception center for the city of Japan that you're in and scheduling a date, purchasing a sticker or ticket also called a sodai gomi ticket for the required fee from a convenience store or supermarket and placing the bed at a designated spot on the correct morning with the ticket attached to the bed or other furniture you want to dispose of.
Some people in Japan have cut up a mattress and disposed of it bit y bit, although it can take months to throw the mattress or bedding away bit by bit.
So the easier and faster way of disposing of the mattress in Japan or bedding in Japan is to schedule a pickup, through the bulky waste pickup.