Firstly, I'm pretty sure if you read it again you'll find there isn't any FTL (faster than light) travel in Ender's Game, they do travel at significant portions of light speed in the novel but don't exceed it.
That aside, in response to the question.
Short answer: Yes.
Longer answer: Those on the ship traveling at very very fast speeds would appear to be moving (& speaking) in slow motion on the earth-side end of the conversation / conversely the people on earth they're talking to will appear to those on the speeding ship to be moving & speaking at high speed.
The exact speed would depend on what fraction of light speed the ship is travelling at, if the ship is travelling at half light speed then people on earth appear to be moving at double speed to those on the ship & those on the ship appear to be moving at half speed to those on earth, the closer to light speed it goes the faster those on earth appear to move & the slower those on the ship seem to be.
So if the ship isn't travelling at very large percents of light speed you probably don't need to make any adjustments, slow your tape player (or more modern device for playing recorded sound) to half speed or speed it up to double speed & it might sound funny but you can still understand the words.
You could get around it by using ordinary technology to speed & slow the feed at the respective ends, then you do get an artificial lag on the earth-side of the conversation (& none on the ship).
So in summary: Yes (at near light speed) you are going to need to slow down the message as it's received on the ship to be able to understand it, but at slower speeds of merely a significant portion of light speed (say 50%) you probably wouldn't need to.
Side note: FTL communications by the way is a literary device that's as equally impossible as FTL travel, it often relies on properties of quantum entanglement that (to put it bluntly) simply don't exist.
Further side note: Orson may have taken a different tack on things though.