No, YouMail does not get rid of your current voicemail messages. This does not apply to your carrier though. Often, if inboxes aren't being used, they may be cleared out for data storage reasons. You may want to check with them on their message retention policy.
YouMail begins working once you activate and from that point forward, your messages are handled by us. Anything up to that point is handled by your carrier.
If you would like to retrieve your old voicemails, check out this link for more info.