Nope. If you plan on only making outgoing calls and do not expect anyone will need to call your Twilio phone number back, you do not need to purchase a Twilio phone number.
You can use the REST API to make outgoing calls from your account using any of your verified phone numbers as the "From" number (caller ID). You can add a verified phone number by clicking the "Verify a Number" button in your account "Numbers" page.
In the event that someone decides to call you back, the call will be made to the phone number you used as your caller ID.