If the quest should be completed in certain time, you will see a timer in the quest window. It can be of two types:
Yellow timer means that if you fail this quest, you can start it again for free
Red timer means that this quest cannot be restarted if you fail, although you can skip it using superbucks
When do event timers start
This is important: timed events start the moment you launch the game, not when you begin collecting items for it.
Difference between event timers and quest timers
Almost all events are timed, but some events also have timed quests inside them. This means that there will be two different kinds of timers: one for the overall event and individual timers for each stage of the quest.
Please do not confuse the two kinds of timers: the one for the overall quest is located on the right side of the screen, under the event icon, and the individual timer for each stage is displayed in the upper left corner of the quest window.