Praise be to Allah.
Allah, may He be glorified in exalted, created man in a beautiful form, in proportion and in a manner that is suited to this life, so that he will be able to carry out his duties to Allah both outwardly and inwardly.
Allah, may He be exalted, says (interpretation of the meaning): “We have certainly created man in the best of stature” [at-Teen 95:4].
Hence Allah, may He be exalted, forbade man to change his appearance for the sake of beautification, because this is like objecting to what Allah has created, and suggests that man thinks that he can do better than what Allah did.
But if this change is not for the purpose of beautification; rather it is to remove a defect that is present, or to ward off some harm that is feared, then in that case making this change is permissible and there is nothing wrong with it.
Please see the answer to question no. 129370.
It was narrated that ‘Abdullah ibn Mas‘ood (may Allah be pleased with him) said: May Allah curse the one who does tattoos and the one who has a tattoo done, the one who plucks eyebrows and the one who has her eyebrows plucked, and those who file teeth for the purpose of beautification, changing the creation of Allah… Why should I not curse those whom the Messenger of Allah (blessings and peace of Allah be upon him) cursed, when it is in the Book of Allah? … “And whatever the Messenger has given you – take; and what he has forbidden you – refrain from” [al-Hashr 59:7].  
Narrated by al-Bukhaari (5931) and Muslim (2125).
An-Nawawi (may Allah have mercy on him) said:
With regard to the words “Those who file teeth for the purpose of beautification,” this indicates that what is haraam is that which is done for the purpose of beautification. But if there is a need to do that for medical reasons, or to correct a defect in the teeth and so on, then there is nothing wrong with it. And Allah knows best.
End quote from Sharh Saheeh Muslim (14/106-107).
If this birthmark is in a form that is causing harm to you, such as if it is off-putting to the one who sees it, and you are married or about to get married, and you are afraid that your wife would be put off by you because of it, then in this case there is nothing wrong with removing it.
Shaykh ibn ‘Uthaymeen (may Allah have mercy on him) was asked:
There is a girl who has small black spots like moles on her face; there are several of them, between six and eight spots, scattered on her face. She is asking: What is the ruling on having things like these spots removed in the hospital, using a laser or some other method?
He (may Allah have mercy on him) replied: There is nothing wrong with removing them, because when there are as many as you have mentioned, it is undoubtedly disfiguring to the face, and is likely to be off-putting to people. The basic principle here is that whatever is done for the purpose of beautification, then it is haraam, but whatever is done to remove a defect, then it is halaal. The evidence for the former is that the Prophet (blessings and peace of Allah be upon him) cursed the woman who does tattoos and the woman who has them done, and the woman who files teeth and the woman who has that done, because these things are done for the purpose of beautification.
The evidence for the second, which is removal of defects, is the fact that the Prophet (blessings and peace of Allah be upon him) gave permission to a man whose nose had been cut off to acquire a nose made of silver, but it began to stink, so he instructed him to acquire a nose made of gold, instead of silver, because this comes under the heading of removing a defect.
So you may follow this principle and benefit from it. Based on that, we may say that the attempts of those women who try to change the dark colour of their skin and make it lighter are haraam, because this comes under the heading of changing the creation of Allah merely for the purpose of beautification.
If someone were to ask: What do you say about correcting strabismus (nonaligned eyes)? Does this come under the heading of beautification or of removing a defect?
Answer: This comes under the heading of correcting a defect, so it is permissible. Similarly, if the teeth are clearly protruding, this is regarded as a defect, so there is nothing wrong with straightening them and making them level with the other teeth.
End quote from Fataawa Noor ‘ala ad-Darb (11/83).
If there is the fear that this birthmark may lead to some kind of skin disease, as has been stated by some doctors, then there is a stronger reason for the concession allowing removal of it.
Hence we advise you to consult a trustworthy dermatologist.
And Allah knows best.