The removal of the frenum of the lips is called frenectomy. This surgical procedure is mostly performed in the upper lips. A frenum that is attached to the top of the alveolar crest could cause diastema and orthodontic problems. That is why a frenectomy should be performed. In these cases it is called a labial frenectomy.
When the frenum is short it puts too much tension to the gingiva and should be removed. Here is a short video describing the frenectomy dental intervention: