How to remove item from an array in PowerShell?

The best way to do this is to use Where-Object to perform the filtering and use the returned array.

You can also use @splat to pass multiple parameters to a command (new in V2). If you cannot upgrade (and you should if at all possible, then just collect the output from Get-ChildItems (only repeating that one CmdLet) and do all the filtering in common code).

The working part of your script becomes:

$moreArgs = @{}
if (-not $NoRecurse) {
  $moreArgs["Recurse"] = $true
}

$filesToDelete = Get-ChildItem $BackupDir @moreArgs |
                 where-object {-not $_.PsIsContainer -and 
                               $_.LastWriteTime -lt $(Get-Date).AddDays($days) -and
                              -not $_.FullName.Contains($exclusion)}

In PSH arrays are immutable, you cannot modify them, but it very easy to create a new one (operators like += on arrays actually create a new array and return that).

I agree with Richard, that Where-Object should be used here. However, it's harder to read. What I would propose:

# get $filesToDelete and #exclusionList. In V2 use splatting as proposed by Richard.

$res = $filesToDelete | % {
    $file = $_
    $isExcluded = ($exclusionList | % { $file.FullName.Contains($_) } )
    if (!$isExcluded) { 
        $file
    }
}

#the  files are in $res

Also note that generally it is not possible to iterate over a collection and change it. You would get an exception.

$a = New-Object System.Collections.ArrayList
$a.AddRange((1,2,3))
foreach($item in $a) { $a.Add($item*$item) }

An error occurred while enumerating through a collection:
At line:1 char:8
+ foreach <<<< ($item in $a) { $a.Add($item*$item) }
    + CategoryInfo          : InvalidOperation: (System.Collecti...numeratorSimple:ArrayListEnumeratorSimple) [], RuntimeException
    + FullyQualifiedErrorId : BadEnumeration