On Star Wars Episode VI: Return of the Jedi we see Darth Vader dying next to an Imperial shuttle as he speaks with Luke. Why didn't Luke use the Force to move Vader onto the shuttle? I assume he was capable of this since he moved C-3PO around with the Force earlier in the film.
Either Luke did use the Force to move Vader's body onto the shuttle, or he didn't need to. (Meaning he was physically strong enough to get Vader's body up the ramp and onto the shuttle.) Whichever way it happened, Luke obviously was able to get Vader's body onto the shuttle.
Note that in the scene on the Death Star II it wasn't Luke that stopped them from getting on the shuttle, it was Vader. He knew his death was imminent, and he wanted to properly say goodbye to Luke and to see him face to face for the first (and only) time. If Vader gets on the shuttle, there's a chance he dies before they land on the Forest Moon, and he never has a chance to say goodbye to Luke. (And to ask Luke to tell Leia that there was still some good in him.)
Luke also manages to get Vader's body off the shuttle and onto the funeral pyre he built. Again either he uses the Force or he doesn't need to. (My bet is on Luke at least using the Force to enhance his strength, since Vader's pretty big compared to him.)
There’s two components to this as far as I can see:
Dramatic license - it’s the writer’s privilege to be able to pick and choose what happens when in order to drive events or add weight to them. In this case seeing Luke nonchalantly skipping along while he casually floated his dad alongside him wouldn’t have been very dramatic.
Unlike the sequel trilogy where the Force is portrayed as a toolkit of easily accessed tricks, even for the inexperienced, in the original trilogy every time we see Luke use the Force for substantive effect (think: light sabre retrieval in the Hoth ice cave and, as you point out, levitating 3-PO) we see him enter a state of meditation. True, we see him perform a light sabre retrieval with less focus and effort in Jedi, but this is consistent with his training on Dagobah and further experience with the technique, as compared to that moment on Hoth. The circumstances around his and Vader’s departure from Death Star II were hardly conducive to such meditation, and we’ve never (up to this point) seen Luke able to use Force levitation without entering that calm, focussed, meditative state, let alone whilst engaged in some other activity, even as basic as walking around.
Ultimately however, since it is never explained (i.e. portrayed) how or why he does or does not levitate Vader into the shuttle, any explanation either way can only ever be a subjective opinion.
It is my personal opinion (other opinions are available and just as valid) that what we see in Return of the Jedi is internally consistent with the portrayal of Lukes use of the Force in the previous 2 episodes and that this question only arises because of the INconsistency as now portrayed in episodes 7-9 (And to a lesser extent in episodes 1-3), where-in the Force is portrayed as nothing more problematic to wield than a trusty blaster at your side (altho at least in the case of the prequels this is arguably explained by the Force being wielded by fully trained and experienced Jedi).
Easiest answer: because it's a film, and it's far more dramatic a moment to show him struggling to drag his father than it would be to have him use the Force. Mind, they could have made his usage of the Force just as much of an exertion, but there's something more personal and tactile about him physically dragging Vader. It's an experience we, as an audience, can directly related to.
According to the d6 Star Wars RPG (REUP edition, p163)
Telekinesis Alter Difficulty: Very Easy for objects weighing one kilogram or less; Easy for objects weighing one to ten kilograms; Moderate for objects 11 to 100 kilograms; Difficult for 101 kilograms to one metric ton; Very Difficult for 1,001 kilograms to ten metric tons; Heroic for objects weighing 10,001 kilograms to 100 metric tons.
Object may be moved at 10 meters per round; add +5 per additional 10 meters per round. The target must be in sight of the Jedi.
Increased difficulty if object isn’t moving in simple, straight- line movement: + 1 to +5 for gentle turns. + 6 to +10 for easy maneuvers. +11 to +25 or more for complex maneuvers, such as using a levitated lightsaber to attack. Modified by proximity. This power may be kept “up”
This would put Luke at a Moderate difficulty (11-15, Vader is 88.9kg) plus a modifier of +6 to +10 for maneuvering, he would likely be at a +2 (close but not touching) for proximity, and +5 if he's rushing (space station blowing up). As pointed out by others, he's been wounded, I would say "wounded twice" giving him a penalty of -2D (Luke has a 10D+2 during RotJ).
All said, Luke needs to roll a 24-28 on 8D+2. Assuming Vader is not resisting and wanting his son to get to safety. If he was, add Vader's Perception (3D+1) or Control (11D+1) roll to the difficulty.