Disabling the voicemail service on your iPhone is fairly simple and can save you a lot of money. This is particularly true when travelling internationally; routing calls through other service providers is relatively costly. In addition to disabling your voicemail through your phone, you can also do it by calling your service provider's customer service line or by logging into your account online (where you can also adjust your other features).
Turn on your iPhone.
Punch in "#004#" (without the quotation marks)
Press "Call." This will disable your voicemail.
You can reactivate your voicemail by punching in "004#" and pressing "Call."