True or false the earth loses energy to space only at night

Well first of all, there's nothing that "the Earth" does at night,
because 'the Earth' is never completely 'night'.   "The Earth"
is always half night and half day.

Now, your question probably means:  The Earth loses energy
to space only from the part that's dark, where the sun don't shine.

That's false.  The Earth is warm and space is cold, so there's
heat radiating into space from every place on Earth all the time.
Fortunately for us, the sun shines on every place on Earth for
50% of the time, and pours down as much energy as radiates
away into space.

And more.  That, and the fact that we've been pumping stuff into
the air for the past 300 years that makes it harder for heat to get
away into space, are the reasons why the Earth is getting hotter.
Can it keep going ?  Sure.  But past a certain point, we can't
live in it, so we start to decrease, and we eventually go extinct.