What are the potential/hypothetical consequences if an exchange does not allow you to manage your own private keys?
If an exchange is managing the private keys (i.e. you are not managing your own private keys), it means that the exchange controls your money. They are then like a bank; you deposited money in and got a promise that they will give you the money later when you ask for it. This means that you are trusting them to hold onto your money securely.
Because they control the private keys, they could steal your money by taking it and running away. Or if the exchange were to be hacked, the money could be stolen by hackers. Both of these scenarios have occurred in the past. In either case, your money would be gone and there would not be much that you could do about it.
Are there any exchanges that allow this option?
In the traditional sense of an exchange where you have an account and login (e.g. Bitstamp, kraken, Bitfinex, etc.), no. No exchange allows that for multiple reasons: controlling your own private keys means that during an actual trade, you may scam the other trader by taking their money and not releasing the amount you were to pay; it also means that the exchange will work less efficiently as instead of doing trades in an internal database, trades must occur on chain.
There are decentralized exchanges which allow you to control your own private keys such as Bisq. But few people use them and there is a bit more risk involved because the other party could scam you.