It is illegal to dump furniture, mattresses or fridges, etc. in the street hoping that someone will come along and take it away.
If you are caught ‘fly tipping’, as it is called,* you can be fined. More information here.
However, the council realises that not everyone can transport these things and, if you phone them, they will make an appointment to collect them. It costs £16. You agree a day when you will put out the things to be collected and it needs to be ready and waiting for them by 6.00 AM that morning.
See Southwark Council’s bulky collection web page for more details of what they will – and will not – collect.
* To do something ‘on the fly’ means to do it on the run or in a hurry.