I am trying to get my company to transition to Microsoft lync. Problem is, a few employees share the same computer. Can two computer users share Microsoft Lync or will each user have to log off when they are done?

Either.  Are you planning to use it just for IM/P or for voice?  Are the users using the same computer using the same user account on the computer as well?

They could use a shared Lync account, or sign in to Lync using their own credentials even if the computer is signed in as another user if they're like.