West Berlin had plenty of marshy and other marginal areas that could be used for a landfill, especially in an era when incineration of solid waste substantially reduced the volume (and helped reduce the need for coal).

Sewer pipes continued to run to treatment facilities in East Berlin. Much later, in the era of the Wall, the pipes were barred to prevent escapes, but West Berlin continued to send sewage east and pay the East for its treatment.

For clarification:

What type of waste are you specifically asking about? This may help you get your answer. There are various types of waste, for example wastewater, which for obvious reasons cannot be flown out on planes.