He wants to close a gap between his teeth
The basic principle with regard to cosmetic procedures that are done to remove a fault or deformity are that they are permissible, because Islam is a religion of ease that aims to make things easy for people.
It was narrated that ‘Arfajah ibn As’ad (may Allaah be pleased with him) said: My nose was cut off on the day of al-Kulaab during the Jaahiliyyah (a famous battle) and I had a nose made of silver, but it began to smell bad, so the Messenger of Allaah (peace and blessings of Allaah be upon him) told me to get a nose made of gold. Narrated by al-Tirmidhi (1770)’ classed as hasan by al-Nawawi in al-Majmoo’ (1/245)and by al-Albaani in Saheeh al-Tirmidhi.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
Beautification is of two types: beautification that is aimed at removing a fault due to an accident etc – there is nothing wrong with this, because the Prophet (peace and blessings of Allaah be upon him) told a man whose nose had been cut off in battle to get a nose made of gold.
The second type is aimed at increasing beauty, which is not aimed at removing a fault, rather the aim is to make someone more beautiful. This is haraam and is not permissible because the Messenger (peace and blessings of Allaah be upon him) cursed the woman who plucks eyebrows, the one who has that done, the woman who does tattoos and the woman who has that done, the woman who adds hair extensions and the woman who has that done, because this is aimed at increasing beauty and is not removing a fault.
Fatawa ‘Ulama’ al-Balad al-Haraam (p. 575).
A large gap between the teeth is an abnormality which some people are affected by. It is permissible for one who has such a fault in his teeth, or other faults, to correct it, and there is no sin on him for doing that, in sha Allaah.
Al-Nawawi (may Allaah have mercy on him) said in Sharh Muslim (13/107):
With regard to the phrase “women who file teeth”, what is meant is doing that for the purpose of beautification. This indicates that that which is haraam is that which is done for the purpose of beautification. But if there is a need for that for medical reasons or to correct a fault in the teeth etc, there is nothing wrong with that. And Allaah knows best. End quote.
The Standing Committee for Issuing Fatwas (24/75) was asked: If I have two teeth that are long, is it permissible for me to make them level with the other teeth?
If they are so long that they bother you, they you may remove that which is bothering you only. End quote.
And Allaah knows best.