Demonstrates how to relink images using Sitecore PowerShell Extensions. Comment included by Dylan shows how to Remove the link and remove the item.

Relink Sitecore Image.ps1

$item = Get-Item -Path "master:\media library\images\koala"
$itemNew = Get-Item -Path "master:\media library\images\penguins"
$links = Get-ItemReferrer -Item $item -ItemLink
foreach($link in $links) {
    $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
    $itemField = $linkedItem.Fields[$link.SourceFieldID]
    $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)
    
    $linkedItem.Editing.BeginEdit()
    $field.Relink($link, $itemNew)
    $linkedItem.Editing.EndEdit() | Out-Null
}

Remove Link and Item.ps1

# Crafted by Dylan

function Remove-ItemLink {
    param([Item]$item)
    
    $linkDb = [Sitecore.Globals]::LinkDatabase

    $links = Get-ItemReferrer -Item $item -ItemLink

    foreach($link in $links) {
        $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
        $itemField = $linkedItem.Fields[$link.SourceFieldID]
        $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)

        $linkedItem.Editing.BeginEdit()
        $field.RemoveLink($link)
        $linkedItem.Editing.EndEdit()
    }
}

# Example usage: delete items along with their references that have passed a certain date defined by a 'date' field
$today = Get-Date
$todayIsoDate = [Sitecore.DateUtil]::ToIsoDate($today)
$itemsToDelete = Get-Item -Path master: -Query "/sitecore/system/Modules/Mysite/Service Schedules/*[@date < '$($todayIsoDate)']"

Foreach($item in $itemsToDelete) {
    Write-Host "Cleaning up $($itemsToDelete.Paths.Path)"
    Remove-ItemLink -Item $item
    Remove-Item -Path $item.Paths.Path
}

I think it would be super useful if we added a couple of commandlets, based on that brilliant bit of code
like Get-ItemReferrers & possibly Relink-Item. You did it again! I didn't think it would be either that succinct or even viable!

Also - great use case scenario!

Updated to use Get-ItemReferrer and Out-Null.

Great script! Can I suggest this so that it works in multilingual scenarios:

$linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID -Language $link.SourceItemLanguage

Just found this gist incredibly useful, along with this stack overflow post: https://sitecore.stackexchange.com/questions/1297/removing-all-references-to-an-item-using-ootb-tools/1307

It has resulted in this function that I'm using:

function Remove-ItemLink {
    param([Item]$item)
    
    $linkDb = [Sitecore.Globals]::LinkDatabase

    $links = Get-ItemReferrer -Item $item -ItemLink

    foreach($link in $links) {
        $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
        $itemField = $linkedItem.Fields[$link.SourceFieldID]
        $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)

        $linkedItem.Editing.BeginEdit()
        $field.RemoveLink($link)
        $linkedItem.Editing.EndEdit()
    }
}

# Example usage: delete items along with their references that have passed a certain date defined by a 'date' field
$today = Get-Date
$todayIsoDate = [Sitecore.DateUtil]::ToIsoDate($today)
$itemsToDelete = Get-Item -Path master: -Query "/sitecore/system/Modules/Mysite/Service Schedules/*[@date < '$($todayIsoDate)']"

Foreach($item in $itemsToDelete) {
    Write-Host "Cleaning up $($itemsToDelete.Paths.Path)"
    Remove-ItemLink -Item $item
    Remove-Item -Path $item.Paths.Path
}

Thanks for sharing @dylanmccurry-sage!