Supposedly Miyamoto was quoted as saying the film influenced the original game but I'm thinking that was a misquote. Some posters on IMDB have said he was quoted in the early 90's as saying he simply saw the film. That seems much more likely but I can't confirm that either. The Hylia website addressed the issue in the article Zelda: Based on a "Legend"? but I don't know if it's still there. There are also two similarities that I can think of off the top of my head which couldn't have been anything but a coincidence.
Miyamoto has said he got the inspiration for Zelda by exploring a cave as a kid. He was afraid to go in the cave until after he found a lantern. A few of the Zelda games including Twilight Princess have a lantern or candle as an item.
Jack used a lantern to look for Lily in the snow.
In Ocarina, the Kokiri children couldn't leave their forest or they would die.
On the 2002 director's cut commentary, Ridley said a possible reason for Jack not running off with Lily like the US is that as a forest boy, he probably can't leave the forest. Anyone know, by the way, where this idea ultimately originates from?