You can add a variable in release variables Tab, and then use PowerShell task to update the content of your settings.json.
Assume the original content is
{
test : "old
}
And you want to change it to
{
test : "new"
}
So you can replace the variable in json file with below steps:
1. Add variable
Define a variable in release variable tab with the value you want to replace with (variable test with value new
):

2. Add PowerShell task
Settings for powershell task:
Type: Inline Script.
Inline Script:
# System.DefaultWorkingDirectory is the path like C:\_work\r1\a, so you need specify where your appsettings.json is.
$path="$(System.DefaultWorkingDirectory)\buildName\drop\WebApplication1\src\WebApplication1\appsettings.json"
(Get-Content $path) -replace "old",$(test) | out-file $path