No, it does not. AnyMem is not configured to remove Trados templates or interfere with your Trados in any way. However, you cannot work both with Trados and with AnyMem simultaneously.
The interaction between AnyMem and Trados is as follows:
1. Upon MS Word startup, both AnyMem and Trados templates are active.
2. The first AnyMem operation (like the Start Translation command) disables Trados templates until next time you run MS Word.
To resume using Trados templates:
1. Click the "Finish translation" command on the AnyMem toolbar
2. Close MS Word and run it again