My spider is moulting, but hasn’t moved for ages.

If it’s an adult this process could take up to 24 hours.  Touching it could do more harm than good.

My spider has completed its moult, but the old skin is still stuck to the abdomen.

Dampen the skin with a few drops of water and allow this to soak in for a few minutes.  Grab the skin with forceps and let the spider pull itself free (don’t pull yourself).  You could add a few drops of glycerine to the water drops, but avoid this if at all possible, because it will damage the tarantula’s natural waterproofing.  If this glycerine water enters the book lungs it could kill the spider.  

My spider is definitely stuck in its moult.

If the spider has a couple of legs struck in the moult you can get the spider to shed them by grabbing the offending leg’s femur and tugging it.  It is best to get the spider to shed trapped legs, particularly if they have begun to harden.  If the abdomen is stuck, see above, but if the book lungs have not been shed your spider will die, no matter what you do.