Saving powershell output not working

You don't need the $output variable at all.

Also, the Write-Host cmdlet sends its output directly to the host, not to the "Success" stream (which is where you want it)

Get-content "C:\Users\Admin\Desktop\Sites\prss.txt" |
    ForEach-Object {
        if (Test-Connection -ComputerName $_ -Count 1 -ErrorAction SilentlyContinue){
            Write-Host "$name is up" -ForegroundColor Green
            "$_ is up"
        } else {
            Write-Host "$name is down" -ForegroundColor Red
            "$_ is down"
        }
    } | Out-File  'C:\Users\Admin\Desktop\Sites\results.txt'
Start-Sleep -Seconds 15

There is nothing in the $output variable.

$names = Get-content    "C:\Users\Admin\Desktop\Sites\prss.txt"
$output = @()           # define empty array that we will add to
foreach($name in $names){
    if (Test-Connection -ComputerName $name -Count 1 -ErrorAction SilentlyContinue){
        Write-Host "$name is up" -ForegroundColor Green
        $output += "$name is up"
    } else {
        Write-Host "$name is down" -ForegroundColor Red
        $output += "$name is down"
    }
}
$output | Out-File  'C:\Users\Admin\Desktop\Sites\results.txt'
Start-Sleep -Seconds 15

You can share the error you are getting.

Meanwhile do this,

Modify this line.

$output | Out-File 'C:\Users\Admin\Desktop\Sites\results.txt'

To this.

$output | Out-File -FilePath C:\Users\Admin\Desktop\Sites\results.txt