Once a timed quest expires, that's it, it can never be completed in that save file and you'll never see it in your list of quests again. This does indeed mean that you won't be able to get their rewards. For example, there are only two copies of the Lime Bangle equipment available: one from a normal quest, and one from a timed quest, meaning that if the timed quest expires you'll only ever be able to have one instead of both. But you don't need to worry about missing anything completely:
- All timed quest rewards are available from some other non-timed source, even if that source is inconvenient or limited. (Like the Lime Bangle example above, or gems that you otherwise have to craft yourself, or shops that don't exist until later, or rare enemy drops.)
- There will always be enough reputation available to get five stars with every area, even if you miss all its timed quests. (Note that this may require being overlevelled for the story.)
There are some minor exceptions. There are a few timed quests that give you gems of ranks otherwise unavailable. But when you can get Poison Attack III gems, who cares about a unique Poison Attack II gem? Note that this can get pretty complex: Paralysis IV is only available through a timed quest and a mutually-exclusive quest, and you can only get Paralysis V crystals from level 70-90 enemies.
There is also a reasonably major exception. One chain of timed quests that travels through Frontier Village and Alcamoth leads to a final non-timed quest against a Unique monster. You do not get any exclusive rewards, but if you miss a step at any time before it expires, you lose the chance at quite a few subsequent quests.
(Wikia source; contains location spoilers)