Can a Battle Master fighter continue to use their maneuvers after spending all of their superiority dice? Or can they not do any more maneuvers until they finish a short or long rest?
It is not clearly stated, but Superiority dice are required to use Maneuvers.
The rules say:
When you choose this archetype at 3rd level, you learn maneuvers that are fueled by special dice called superiority dice.
Maneuvers tell you to "expend one superiority die" or similar wording as part of the description of the effect, but it does not make it clear that the maneuver is contingent on the expenditure in the wording. This could be better edited. However, the general wording, and the textual reference that says that maneuvers are "fueled" by superiority dice, make the intention fairy clear.
No, they can't use up more maneuvers when they're out of dice. And yes; they have to take a short or long Rest in order to get the dice back to then be able to do maneuvers.
After reaching 3rd level in the Fighter's class and choosing the Battle Master subclass, you're given 4 Superiority Dice to use as maneuvers (You get one more upon reaching level 7 and level 15). These can be used in a variety of interesting and tactical ways, but after expending it you will be forced to wait until your next short or long Rest in order to get these back.
This doesn't sound like a problem to me, in comparison to the Bard's Inspiration Dice. A bard gets a total of their Charisma modifier in dice, which he can use in a few ways, but upon use, the bard will have to wait until a long Rest to get these dice back for the day.
(Keep in mind that the Fighter is only allowed to use one maneuver per attack.)
... under combat superiority it says "maneuvers are fueled" by the special die. BUT in the description of the maneuvers themselves rather that saying " When you hit a creature with a weapon attack, you expend one superiority die to...." It says "When you hit a creature with a weapon attack, you CAN expend one superiority die to...." the word can suggest to me that you do not have to expend a superiority die...
Especially when you look at the parry maneuver that specifically says, "can use reaction and expand one superiority dice" which indicates for this to work you must use a superiority dice. Same wording structure for Rally Maneuver as well as riposte