This would certainly have earthshattering implications for mathematics. One of the deeply rooted concepts within mathematics is the idea of the natural numbers (0, 1, 2, 3...). If we were to assign numbers to each question, we would see that there's always a next question as long as there's at least a mapping from natural numbers to questions.
A trivial mapping would be "What is the most interesting feature of the number X" where X is a natural number.
If we ran out of questions, that would suggest that something fell apart in the mathematics of natural numbers, because natural numbers should always be able to yield one more question. That would be quite earthshattering... and most likely spawn its own questions.
Now there are other ways we could run out of questions, but they do need to be addressed on a case by case basis. For example, if we run out of questions because we ran out of energy in the universe with which to ask them, that calls for a very different sort of contingency plan.