IRL the relativistic effects remain so long as you're actually accelerating through spacetime. Something like the Alcubierre warp drive (assuming it's possible to construct, which it probably isn't) would avoid relativistic issues because it would actually warp space around the ship, the ship itself wouldn't move, so there's no acceleration and ergo no time dilation.
Mass Effect is fictional, so it doesn't have to follow the IRL rules, and they kind of hand-wave it. Eezo allows you to create "mass effect" fields which alter the laws of physics and allow you to get from point A to point B sooner than a light beam would without actually travelling through space at faster than the speed of light. You're moving a bubble of low-mass space around, but the ship itself isn't really moving in the conventional sense. This is how you can travel from one system to another and back faster than light without arriving before you departed.
The relays create narrow pathways of this low-mass space between themselves using mass effect fields way more powerful than what the citadel races can create, allowing ships to move ultra-fast between relays. While the citadel races can't duplicate the relays, they were able to create a comm buoy system that uses the same principle to create similar pathways big enough to send a laser down, which is how they send messages from one system to another faster than light.
From what I recall the codex entries never explicitly mention time travel, but the fact that you don't have relativistic effects from FTL in the mass effect universe is implicit in the fact that throughout the games you hop all over the galaxy but it never seems to create any problems with linear time.
@dean: I had forgotten about the QEC. That was supposedly something super secret and brand new developed by Cerberus, so they're probably just going to pretend it doesn't exist.
As far as the end wave expanding faster than the mass relays, I just assumed it was traveling at the speed of light, so when it ended up in the low mass space between relays it was travelling at the speed of light in that low-mass space, which is much much higher than the regular speed of light. Even though the Normandy was also travelling through that space it was going slower than the speed of light in that space and so was overtaken.