Praise be to Allah.
If the teeth are far apart in a such way that causes embarrassment to the person, there is nothing wrong with closing the gaps by means of what is known as a bridge. But if that is only for the purpose of beautification and enhancing beauty, without there being any defect or deformity in the teeth, then this is the type of beautification that is not allowed, because of the report narrated in as-Saheehayn from ‘Abdullah ibn Mas‘ood (may Allah be pleased with him), who said: May Allah curse the woman who does tattoos and the woman who has tattoos done, the women who pluck their eyebrows and the women who file their teeth for the purpose of beautification, changing the creation of Allah.” Narrated by al-Bukhaari (4507) and Muslim (3966).
Al-Qurtubi (may Allah have mercy on him) said: The hadiths testify that the one who does any of these things is cursed, and that they are major sins. But there is a difference of scholarly opinion regarding the reason for which they were prohibited. It was said that it is because they come under the heading of deceit, or because they come under the heading of changing the creation of Allah, may He be exalted, as Ibn Mas‘ood said. The latter is more correct, and also includes the first reason. Moreover, it was said that what is prohibited is only that which endures, because it comes under the heading of changing the creation of Allah, may He be exalted. As for that which does not endure, such as kohl and other things with which women adorn themselves, the scholars have permitted that."(Tafseer al-Qurtubi 5/393).
An-Nawawi (may Allah have mercy on him) said in Sharh Muslim (13/107): With regard to the words “the women who file their teeth for the purpose of beautification,” what is meant is that they do that for the purpose of making the teeth appear beautiful. This indicates that what is haraam is that which is done for the purpose of beautification; but if it is needed for the purpose of treatment or to correct a defect in the teeth and the like, there is nothing wrong with it. And Allah knows best. End quote.
Please see also the answer to question no. 82409 .