J.R.R. Tolkien’s invented language based chiefly on Finnish was Quenya.
Quenya is one of Tolkien’s best-developed Elvish languages. Its sound and structure were strongly influenced by Finnish, while its vocabulary and history were created within Tolkien’s own linguistic mythology. In the legendarium, Quenya was associated especially with the High Elves and with the Elves who had lived in the West.
Tolkien did not treat his invented languages as simple collections of replacement words. He built historical relationships, sound changes, grammatical systems, and writing traditions for them. This philological approach helped make Middle-earth feel like a world with deep cultural history.
Sindarin is another major Elvish language, influenced in part by Welsh and used widely among Elves in Middle-earth. Khuzdul is the secret language of the Dwarves, while Adûnaic is associated with the Númenóreans. Those distinctions make Quenya the specific answer to the Finnish-influence question.