What protocol are you using?
Can you post a session log file?
Though I do not think it's WinSCP that locks the file. Did you try to turn off antivirus?
Did you try using File.Move instead of File.Copy and File.Delete?

Dim transferOptions = New TransferOptions
transferOptions.ResumeSupport.State = TransferResumeSupportState.Off
transferOptions.PreserveTimestamp = False

transferResult = session.PutFiles(UploadFromFolder + filename.FileName, ClientFolderPath, False, transferOptions)

transferResult.Check()
File.Copy(UploadFromFolder + filename.FileName, BackupFolder + Path.GetFileName(filename.FileName))

File.Delete(UploadFromFolder + filename.FileName)

File.Delete gives error that the file is used by another process and cannot delete. It is just a text file and the transfer somehow locks it I think.