If you prefer to add the web.config as a link in your test project as an explicit 'reminder' for being the configuration for your test project, you can put
copy "$(TargetDir)\Web.config" "$(TargetDir)\App.config"
in the Post-build event command line part in Visual Studio located in the project properties under the Build Events tab.
By doing so, the following will automatically be added to the .csproj file of your test proejct:
<PropertyGroup>
<PostBuildEvent>copy "$(TargetDir)\Web.config" "$(TargetDir)\App.config"</PostBuildEvent>
</PropertyGroup>
For this to work, the linked file in the test project needs to have a meaningful value - i.e. Copy always in Copy to Output Directory.
Another approach, like suggested in one of the comments, would be to use the CopyToOutputDirectory directive. One might argue though, that this gives any future maintainer less clues in regards to what is happening during the build process - without having to dig through the .csproj in the test project, that is.
<ItemGroup>
<None Include="..\MyProject\Web.config">
<Link>$(TargetFileName).config</Link>
<CopyToOutputDirectory>PreserveNewest</CopyToOutputDirectory>
</None>
</ItemGroup>