Rules as written states that no, magic items will not work.
"No Mana: No one can use magic at all. Enchantments and permanent
magic effects are suspended and do not function while within a no-mana
zone.". -- Magic 6, Mana Sidebar (Emphasis mine)
Magical items are rendered inert, so nothing magical can ever work in a no mana zone. If the item has any magical properties at all, they are suspended. Even if that object cannot cast a spell directly (or have any direct ambient effect) and can only be used for making potions, it is rendered inert. This is because Alchemy is considered magic ("Elixers are magical in nature" Magic 212).
Naturally magical objects
"Natural" magical objects, such as dragon's blood or a unicorn's horn would be rendered inert in a no-mana zone. If the objects was brought from another dimension or outside area they would no longer be magical when they arrived in the no-mana zone. Of course, if they were brought out of the no-mana zone they would once again be magical as they were simply being "suspended" (see above quote.)
Creating and Using Magic Items
It should be noted that since magic does not exist on this world that enchantment spells cannot be cast, and thus all "crafted" magic items would need to come from "the outside". Anyone using such an item would be unable to activate it or use its ambient ability (in the case of Boots of Haste and a +1 DR piece of armor, respectively).
Alchemy seems like a different beast but is not. Alchemy cannot be learned in a no-mana area (Magic 210). Elixirs are called out as "being magical in nature" (Magic 212). This is strengthened by the fact that characters with magic resistance gain this resistance vs alchemy (Magic 213).
Magically Dependent Beings
In addition, magically dependent beings (beings with the actual Magic Dependency disadvantage) would suffer greatly if they came to that world. Faeries, for example, would immediately begin to diminish if they happened upon this world. (Dependency, B130).