I hope the Rav had a nice shabbos!
If diapers are locked in one’s trunk and one has an old car with no lights on in the trunk. Can one use his car key which is the same car key for his trunk on shabbos? Nothing electrical is happening and it is no different than opening any other door. Is the car muktza machmas chisoron kis? IF not, can one close the car trunk afterword?
It is permitted to open the trunk to get the diapers, provided of course that the area has a reliable eruv.
Concerning closing the trunk, it is certainly permitted if there is a permitted item inside that needs to be looked after by closing the trunk. If there is no permitted item inside, some prohibit closing the door, yet there is some room for leniency where the situation requires it (see sources).
This should preferably not be done in public.
It is forbidden to move even part of a muktzeh item, as we find concerning closing the eyes of the dead (Beis Yosef and Shulchan Aruch 208:42, citing from the Ran). See also Magen Avraham 305:9; Darkei Moshe 311:7; Shulchan Aruch 311:7. The Rema (308:3) likewise states that any movement is forbidden.
However, a car has two distinct usages. One use is for driving, and the other is for sitting inside or for storing items. Concerning the use of the car for driving, the car is muktzeh. However, for use as storage it is not muktzeh, and it is therefore permitted to open the trunk.
This is similar to a fridge. Although the fridge itself is muktzeh (machmas chisaron kis), and it is forbidden to move it on Shabbos, it is nonetheless permitted to open the door of the fridge. The reason for this is that concerning the use of the fridge the fridge is not muktzeh.
The same idea applies to a car. Although it is not permitted to move the car, it is permitted to open its doors or trunk (provided of course that no light goes on).
I heard this idea in the name of Rav Yashiv Ber of Brisk.
Another point is that it is not clear that a car is muktzeh machmas chisaron kis, and it is possible that the car is only a keli she-melachto le-issur, in which case it is permitted to open the doors for a legitimate need (of taking something out). This is the approach taken by the Shemiras Shabbos Kehilchasa (15:25; 20:80) concerning the door to a washing machine, and also concerning a car door.
The difference between the two approaches is closing the car door afterwards. According to the first approach, the door to the car or trunk is simply not muktzeh, and it can be closed as usual. According to the second approach, it is only permitted to close the door of the car or trunk for a need of something permitted, for instance to ensure the safekeeping of permitted items in the trunk – but not for the purpose of looking after the car itself.
This should not be done in front of an am-ha’aretz, who will not understand the distinction, and in general one should avoid doing this in public because it looks strange.