That would be a silly objection, like claiming restaurants are exploiting people's hunger by selling food for money.
That being said, the moment such a platform makes a single iota of data about their users (even seemingly mundane information such as how many users there are, their geographic distribution, how many messages the average user writes etc.) available to anyone for any purpose – including research and non-profit purposes – I would consider the whole enterprise to have turned unethical. It doesn't matter whether the field of psychology could benefit from it, or what the "greater good" might be. It has to be crystal clear and non-negotiable that the only use for those messages is going to be letting other regular users listen to them.
You should also think about what you will do in case law enforcement demands access to user data. The less information you store, the better. Consider automatically deleting messages after a period of time, or similar proactive measures. And of course, maintain a strict process for disposing of backups, else the "delete my data" function is effectively worthless.