Emphasising that a date is near

Solution 1:

When I'm trying to motivate people to pay attention to things like this I usually say that the deadline is looming. It conveys a sense of impending danger; other things that often loom are storms, battles, and obstacles.

You could also say that the deadline is imminent or fast approaching.

Solution 2:

I would probably put something like:

As the deadline for the submission is approaching rapidly, 
we should do without eating and sleeping until then.

The distinction of "approaching rapidly" has a sense of urgency about it, and not specifying the deadline means that people who subconsciously ignore it until it is actually May 25th do not have that option.

And not eating or sleeping for 3 days is probably not the answer to hitting a deadline. Just dead.