does placing a player lower in the lineup save them energy? what about if you DH them?

I think what I'm writing below is correct but don't have strong evidence for it.

1) batting lower in the lineup helps fatigue a very little. Games played and plate appearances seem to both contribute to fatigue. But since you only save an occasional plate appearance, and since plate appearances are only a piece of the fatigue puzzle, it doesn't help very much.
2) DHs fatigue. There appears to be different fatigue at different positions, and DH is one of the slower ones-- but DHing is not a day of rest, just one of slightly slower fatigue.

From my experience over on the Sim League Baseball side, the engine used PA per 162. If you are over the pace for what a player can handle in a full 162-game season, he will begin to fatigue. Thus, DHing doesn't give you the "half day off" some ML managers use it for in real life.
Based on percentages, each spot lower in the line-up will save a player roughly 18 PAs per season over 162 games. For example, odds are such that the lead-off hitter will get roughly 90 more plate appearances than someone 5 spots down in the order over the course of the season.
That being said, your best bet IN THE MINORS (where player development and injury prevention/protecting prospects take priority over winning at all costs) is to bat low durability guys lower in the line-up and change your sac bunt setting to aggressive. -(Sacrifice bunts are not official plate appearances, and will help keep PAs down, as well as prevent massive offensive outbursts where players get 9 ABs against teams running out 0% pitchers).