Since I cant post on the Jagex forums I'm submitting a bug report here.

Area: Quests
Summary: Not being given new quests
Description of Issue: Daily quest timer reset to 24 hours without giving new quests

Steps leading to Issue:
* Logged in via Steam
* I had completed all daily quests earlier in the day
* I had completed the epic quest a couple of days ago; it had 3 days on the reset timer.
* Saw that daily quests had about 4 minutes left before they reset
* Went AFK for a few minutes to wait out timer
* Came back and timers were frozen at about 2:20
* Assumed it was just a visual bug and started a ranked game
* Timers were still frozen after the game
* Restarted the game to see if it would fix the issue
* After relogging the daily quest timers had gone up to 24 hours but all the quests were still marked as completed
* The daily quests were the same as the ones I had yesterday
* The epic quest had properly counted down to 2 days
* Relogged again to see if it fixed the issue; it did not
* Took a screenshot (seen below) showing 23:54:29 on 25 Chapter daily quest, which is not doable

Account Name: AnimaMater on this Steam account
Evidence: http://imgur.com/3qO1aog

EDIT: I should mention that I won the ranked game and the reward screen showed a fresh daily win counter. I am not sure if I started the game before or after the dailies shoud have reset, but I definitely completed the game after the dailies shoud have reset.

EDIT 2: I logged back in to the game after the season end and my quests had reset properly.