You can do whatever you want with your Bitcoin. We will send the BTC amount of your choice to address submitted by you. It is better to be your own wallet, but you can decide to transfer the coins to another exchange.
You're free to transfer your BTC to an address by your choice. It can be an exchange, a wallet of a friend, or something else. BitIreland does NOT keep your funds. They're always sent to an address provided by you (in the order form).