When you archive a contact, they will be removed from your main contact database. They will also be removed from the Global Search Bar auto-match results.
If you are looking for an Archived Contact in MyCase, here are two ways to go about finding them.
Global Search Bar (Everything Results)
Type the person's name into the Global Search Bar on the top-right corner of the screen. Their name WILL NOT appear in the auto-match dropdown results. You will have to click on "Search everything for ____________" on the bottom of the auto-match results. This will take you to the advanced search results screen and the archived contact should appear on the list of items.
Contact Database (Archived Filter)
Open the Contacts tab.
Click the filter called Archived (pictured below).
You will be taken to a list of all of the Archived Contacts in your MyCase account.