I have a sort-of big, green ... plant-like thing in a pot.

I think it's an "Agave Attenuata".

Anyhoo, the ends of its leaves are going brown. Apart from that it seems
pretty healthy. What might be wrong with it? It is currently in a full
shade position - should I move it to where it will get direct sunlight?

Since I have explained the situtation so fully, I expect you will all give
equally detailed answers.

--
Stephen Oakes