How old is the practice of removing wisdom teeth?

Wikipedia is not the only site saying that it started in the 20th or 19th century. Other websites such as The Conversation give similar dates.

"Our grandparents and parents tell stories about the time when kids routinely had their tonsils removed. But for people born in the 1960s and later, their routine surgery stories are about having third molars, a.k.a. wisdom teeth, taken out." (Source: The Conversation)

TL;DR: Extraction of wisdom teeth, also known as third molars or dentes sapientiæ, is documented as early as 1803.

A search of MEDLINE for the MeSH term "Molar, Third" plus text strings "extraction" and "history" yielded 48 documents, of which one seems directly relevant.

• Klein, C., & Lorber, C. G. (1995). Die Entwicklung der operativen Weisheitszahnentfernung [Historical development of surgical wisdom tooth extraction]. Fortschritte Der Kiefer- Und Gesichtschirurgie, 40, 113–116.

I don't seem to have access to this journal, but the English language abstract of the article reads:

Surgical removal of the wisdom teeth is a routine procedure nowadays. Only at the end of the nineteenth century the use of local anesthesia together with the development of radiology led to the establishment of surgical dentistry. Especially the technique of removal of the lower third molar was totally changed and modified many times, depending on the position of the wisdom teeth. First hand instruments were used; later, mechanical devices for bone resection and tooth splitting were employed. Since the 1950s, highly dangerous infections have become rare, thanks to the use of antibiotics. Many publications concerning incision procedure, bone resection and tooth splitting marked the following years.

Looking at related articles yields a couple more possibilities, which I also don't have access to.

• Fukuda K, Endo N, Aoya T, Tsubakimoto M. (1983). [Surgical management in dentistry (2). Management of mandibular impacted wisdom teeth]. Rinshō Shika [Clinical Dentistry], 308, 12-25.
• Schatz JP, Joho JP. (1984). [Preventive extraction of lower wisdom teeth: review of the literature (I)] Medecine et Hygiene, 42, 2285-2286.

From the Klein & Lorber (1995) article abstract, it seems that "end of the nineteenth century" is the answer.

Archive.org has a copy of an 1889 book that contains a few paragraphs about the third molar that mention extraction.

The upper third molar is frequently out of position, projecting toward the cheek and proving a source of constant irritation. In such cases it should be extracted.
The lower third molar frequently fails of complete eruption from lack of room at the angle of the jaw, and in consequence of this the overlying tissue is kept in a state of constant irritation, and often serious inflammations result. In these cases, if either the first or the second molar is much decayed and the third molar is of good quality, extract the first or second, and thus allow the third to move forward and relieve the pressure, otherwise extract the third molar.

• Fillebrown, T. (1889). A text-book of operative dentistry. P. Blakiston, Son & Co.

You may also find interesting this 1926 book that appears to be an 800-page canonical reference on extraction of impacted third molars. The pictures of the instruments in chapter XIV are especially fascinating. Unfortunately, a cursory examination of this book does not reveal a date or period in history when these techniques were developed.

• Winter, G. B. (1926). Principles of exodontia as applied to the impacted mandibular third molar. American Medical Book Company.

In a 1771 book by John Hunter, it is explained that the third molar is also known as Dens Sapientiæ, Latin for "tooth of wisdom." This author mentions extraction of teeth, but not specifically of the wisdom teeth.

• Hunter, J. (1771). The natural history of human teeth: Explaining their structure, use, formation, growth, and diseases. J. Johnson.

One 1726 book explains why the teeth are called "wisdom teeth". This book also mentions extraction of teeth, but primarily for cases of decay.

Dentes, the teeth seldom exceed sixteen in each jaw; the four first in each are called Incisores, the two next Canini, and all the rest Molares; the four last of these are named Dentes Sapientiae, because they do not appear till men arrive at years of discretion.

• Cheselden, W. (1726). The anatomy of the human body. W. Bowyer.

Finally, an 1803 book by Joseph Fox, a surgeon who was a contemporary of Hunter's, specifically mentions extraction of the wisdom teeth. This is the earliest reference to extracting wisdom teeth that I have found so far:

[An improvement to an instrument called the German key] was made for the purpose of fixing a claw, in an advanced position, beyond the bolster, which was found extremely useful in the extraction of the dentes sapientiæ.

• Fox, J. (1803). The natural history of the human teeth, including a particular elucidation of the changes which take place during the second dentition .... Thomas Cox.