When you attempt to close or temp close Fringes of the Eye and summon a barrier with a Check to Defeat: None what happens? Storm was the example I ran into. You can't defeat Storm so you fail the condition to close the location. However is a Check to Defeat: None equal to the barrier being undefeated. This is important because failing to defeat the barrier when attempting to close Fringes of the Eye forces the characters at that location to move to random locations.
Answer and a "that said."
ANSWER: Since Fringes only cares about whether you defeated its summoned barrier, the text triggers: banish Storm, then move each character from Fringes to a random location.
That said... Storm is not "undefeated," nor is a non-check equal to being undefeated. Yes, the distinction can matter.
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Related question: what happens if Fringes summons Pirate Hunting, which wants to (but by rule can't) summon a ship or monster? (Pirate Hunting is defeated or undefeated based on your outcome against the card it summons.)
RAW says "same thing as Storm." It seems a minor overlooking of the rule to let Pirate Hunting bring its "check to defeat" into play despite the chain-summon rule.
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Edit to add only: Perhaps Fringes should read "Summon a random barrier that has a check against it, and defeat it; if you did not defeat it, each character at this location moves to a random location."
If I remember well, it was addressed previously and the bottom line is, if you can't/don't for whatever reason defeat the barrier you draw, then you cannot close the location. Period.
Storm is not defeated when you encounter it, so you do not get to close Fringes of the Eye. A related question came up with Teleportation Chamber and although there was no official response on it when it was brought up there, there was wide consensus that it did not let you close in that case.
Fringes is slightly different since you actually are encountering the barrier as opposed to simply using its checks to defeat, so a "None" doesn't necessarily mean that you automatically fail -- it depends on what the power says. In Storm's case, the power does not give you an ability to defeat it, but other barriers may (Pirate Hunting comes to mind -- note that the check to defeat has been FAQed to "None").
For the other question, by RAW Pirate Hunting would not be able to summon the ship/monster so you wouldn't have any ship/monster to defeat and therefore the location would not close. However, Vic has stated here that you should allow it to summon the card anyway (and extrapolating from that discussion, if the summoned ship was Goblin Weidling, you'd also summon and encounter the Riptide Grindylow).