Steps Involved: 
- Open SharePoint 2010 Management Shell by going to Start | All Programs | SharePoint | Microsoft SharePoint 2010 Products | SharePoint 2010 Management Shell (Run as Administrator). 
- Run the following script.
Powershell Script:
|  ##Remove a link for the specified account name using SharePoint 2010 web service in powershell
 $uri="http://serverName:10736/sites/ECT2/_vti_bin/UserProfileService.asmx?wsdl"
 ## $accountName is a string that contains the account name for which you need to remove a link
 [String]$accountName="domainName\userName"
 ## $accountName is a int that contains the unique ID for the link to remove
 [Int]$id=7
 
 ## Web Service Reference - http://Site/_vti_bin/UserProfileService.asmx
 $userProfileWebServiceReference = New-WebServiceProxy -Uri $uri -UseDefaultCredential
 
 ## To get the user link id please refer the commented script
 # $QuickLinkData=$userProfileWebServiceReference.GetUserLinks($accountName)
 # foreach($link in $QuickLinkData)
 # {
 # Write-host -ForegroundColor Green $link.Name
 # Write-host -ForegroundColor Green $link.Id
 # }
 
 ## Remove the link
 $userProfileWebServiceReference.RemoveLink($accountName,$id)
 |