My leagues have very specific rules about this.
In one redraft league, anyone mathematically eliminated cannot make claims, but only if in order to fill a lineup gap for a late regular season game, they can make a FA pick-up between noon-5 PM on Thursday before games. The idea is that competing teams get access to players 1st, and by that late in the week a FA can be added by a non-contending team so that they put up a fair fight and don't violate our rules about fielding a complete roster.
In my other redraft league, we have a "loser bowl" with a $75 prize among the 8 teams that didn't make the playoffs. Round 1 is 1 vs 8, 2 vs 7, 3 vs 6, 4 vs 5, and round 2 (week 16) is high score takes all among the last 4 teams. Similar to above, teams in the loser bowl may not use the waiver system to make claims, and must wait until Thursday between 10:00 AM - game-time to pick anyone up. This gives the contending teams 1st choice each week of the playoffs, and allows the loser bowl teams to field a competitive team.
So for both of my leagues, non-contending teams may not use the waiver system, but can add FA's once the priority adds are completed. This has worked out ok for us over the years and no one has abused it (e.g. a non-competing team making 5-6 speculative adds) - since it's $4 an add, most teams who aren't competing don't want to accrue that much late season debt, so it's self-policing in a way.