There is no law against destroying banknotes which belong to you, by burning or any other means. (Although I suppose someone who burned their money might come under the Mental Health Act!)
The TV show probably used fake notes. For one thing, real banknotes are quite resistant to fire.
As Arbiter says, defacing banknotes is illegal. This is mainly to avoid people being put in the embarrassing position of receiving banknotes with obscene words, disrespectful alterations to the Queen's picture, etc, on them. This does not apply to destroying them altogether.