Can a character shove a paralyzed person, forcing them to move?
Similarly, could a sorcerer cast the cantrip Gust to force the paralyzed person to move (without a Strength save)?
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The effects you describe do not force the creature to move on its own, but impose an outside force and push it (pulling would work similarly to pushing). These are two separate kinds of "forced movement", the former usually involving mind control or fear effects and the latter involving physical forces, whether summoned, evoked or mundane.
The description of a shove (PHB 195-6, emphasis mine)
If you win the contest, you [..] push it 5 feet
Gust says (XGtE 157, emphasis mine)
must succeed on a Strength saving throw or be pushed up to 5 feet
Contrast this with a spell like dissonant whispers (PHB 234, emphasis mine)
must immediately use its reaction, if available, to move as far as its speed allows
or the "Flee" option of command (PHB 223, emphasis mine)
The target spends its turn moving away from you
A paralyzed creature has a 0 movement speed and no actions available (incl. bonus or reactions) and thus cannot be forced to use them. Even if the spell would grant any of these, the condition would override that effect unless otherwise stated.
Note that the verb choice can be sometimes misleading. The verb "move" can be used both in a transitive (separate actor and subject), as well as an intransitive (actor and subject are the same) way. The above examples use it in an intransitive manner. An example of the transitive use would be the Grasp of Hadar eldritch invocation (XGtE 57, emphasis mine):
you can move that creature in a straight line
In this case the creature performing the move is separate from the one suffering it, and it is thus an outside force that will work on even a paralyzed target.