If you are going to combine strings in this manner you need to wrap them in double quotes, to expand the variables:
$deliveryFolder = "C:\Users\administrator.GTCS\Desktop\Delivery\1.5.6.1140\"
$localFolder = "C:\WebServerGateway\"
$fileSharingServerManifestFileName = "GTCS_GCS_FileSharingServer_Manifest.xml"
Copy-item -Path "$localFolder$fileSharingServerManifestFileName" -Destination "$($deliveryFolder)GcsData\Configurations\Platforms\DEV\Web Server Gateway" -Force
Copy-item -Path "$localFolder$fileSharingServerManifestFileName" -Destination "$($deliveryFolder)GcsData\Configurations\Platforms\INT\Web Server Gateway" -Force
Notice I uses a subexpression $() in the -Destination argument. Because in your case the parser may have trouble determining where the variable name ends, therefore the string may not expand properly.
That said, neither of the arguments are particularly pretty/readable so I'd advise against this syntax. concatenating or better yet using Join-Path to derive the arguments are better choices:
$Path = $localFolder + $fileSharingServerManifestFileName
$Dest = $deliveryFolder + "GcsData\Configurations\Platforms\INT\Web Server Gateway"
Note: Concatenation forces you to pay attention to backslashes separating what would be the -Parent/-Child arguments in Join-Path
Or:
$Path = Join-Path $localFolder $fileSharingServerManifestFileName
$Dest = Join-Path $deliveryFolder "GcsData\Configurations\Platforms\INT\Web Server Gateway"
Then simply modify the commands:
Copy-item -Path $Path -Destination $Dest -Force
Copy-item -Path $Path -Destination $Dest -Force
-Verbose