Olbers Paradox Solution Olbers’ Paradox says that in an infinite universe every line of sight will end on a star. Surface brightness is independent of distance (moving a star further away makes it smaller and reduces its flux but not its surface brightness), so why is the night-sky dark rather than uniformly painted at the brightness of an average star?
Now the explanation I have been given is that the universe is $1.4\cdot10^9$ years old so the furthest we can see is $1.4\cdot10^9$ light years away, but the average distance to a star is $2\cdot10^{24}$ light years away, hence the result.
Now suppose hypothetically that the earth (and sun) are still here the way they are now when the universe is $2\cdot10^{24}$ years old, then will an observer on earth looking at the night sky see uniform brightness? 
 A: 
Olbers’ Paradox says that in an infinite universe every line of sight will end on a star.

That statement is incomplete.  The paradox requires not only an infinite universe, but also one that is both static and infinitely old.  Neither of the second two statements are true for our universe.
Your question considers the effect of aging.  As our position in the universe gets older, light has had more time to reach us and we can "see" a greater volume of space.
But the static condition is not there, so you still don't get a bright sky in the future.  Stars die, and the material to make them is not infinitely abundant.  After a long-enough period of time, there may be no stars to see.  Besides that fact, the universe is expanding faster over time.  The light from distant galaxies may be unable to reach us in the future, limiting the light present.  Our current model of the universe suggests that we will always have dark skies.
A: I agree with what BowlOfRed said, but I'm going to give an answer with a different nuance.
So why is the night-sky dark rather than uniformly painted at the brightness of an average star?
Because the universe isn't infinite. Big bang cosmology describes a 
universe that started small some 13.8 billion years ago and has been expanding ever since. It's been expanding for a finite time, it can't be infinite. 
Unfortunately, in recent years a non-sequitur has crept in wherein a "flat" universe is assumed to be an infinite universe. Articles which used to say the universe was the size of a grapefruit now say the observable universe was the size of a grapefruit. IMHO this is a temporary situation which will be sorted out in a few years once cosmologists appreciate a subtlety of Einstein's greatest blunder wherein a flat universe doesn't have to be infinite, and an infinite universe can't expand. 
By the by, see Expanding Confusion by Davis and Lineweaver: "We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light". And note that the night sky isn't actually dark, it's painted at the brightness of the CMBR which predates all stars.  
A: And still a different answer:  Olber's Paradox fails to take the inverse square law into account.  Light from a star is attenuated by distance.  It is possible for the viewer to be so far away from a star that the star is invisible to the naked eye.  We see that even within our "visible" universe as we record images viewed telescopically that are otherwise invisible to the unaided viewer.  That aspect remains regardless of age or whether the Universe is static or expanding so an infinite Universe would not result in a bright night sky.
