Don't do it.
Some "established character" names have survived in languages and cultures as normal human names. Some are normal human names that just happen to be shared by prominent characters. "Svetlana" is a popular "Russian, Bulgarian, Serbian, Ukrainian and Belarusian name" that enjoyed explosive popularity after it was invented about two centuries ago, despite not being church-approved. It is not inconceivable that Khaleesi, an invented title from George Martin's A Song of Ice and Fire series which became a popular name for girls in 2018, or indeed Beowulf, would be just as ubiquitous in the future.
However, you're writing your book in the present day. Beowulf appears to have never had that Real Name status. The English Wikipedia does not list a single real person by that name.
People named Beowulf probably exist; a casual internet search finds messages by parents who consider naming their children after the character. The problem in your case is exacerbated by your novel being (I assume) a sword-and-sorcery fantasy which is not set on Earth. Out of three fictional characters listed on Wikipedia, one is a version of Beowulf the mythical hero himself, and two others are from sci-fi and of Earth heritage.
If you name your character "Beowulf", most readers will assume you're borrowing and reinterpreting the established character (who is public domain, so this is legal). But if yours is in fact an original character, I strongly advise you to pick another name.
To answer the general question:
What are the ramifications of a character sharing the same name as
another, famous and established character?
Partial matches with existing human names are fine.
With particularly unusual names and wholly fictional worlds, make sure they don't stand out and break immersion. Different Earth readers and reader communities have different standards. It won't bother me if two sword and sorcery characters in the same story are named Reginald and Keiko. For others, these names will only pass muster if there's a Fantasy not-quite-England and Fantasy not-quite-Japan. "Jesus" in fantasy is certainly immersion-breaking for English speakers; I wonder if the same holds true in cultures where it's a popular given name.
Full matches are best avoided; as @wetcircuit points out, this will ruin search engine discoverability. It is also likely to break immersion for readers familiar with the established character.
Additionally, while names cannot be copyrighted, they can be trademarked. A trademark for your character's name belonging to a third party may substantially restrict your opportunities to exploit your work commercially. Ernest Cline's book Ready Player One is largely based on nominative use of real-world intellectual properties, but Warner Bros. Pictures, the makers of the movie, had to license the properties they did not own.