Who can move the Warlock, and how?
DM willing, she can move herself by 'climbing'
As you correctly note, the Levitate spell allows the Warlock to move only by
pushing or pulling against a fixed object or surface within reach (such as a wall or ceiling), which allows it to move as if it were climbing.
If she is tied to a rope, and the other end of the rope is being held by her allied sorcerer, then she can pull on the rope. If the rope is taut she may be able to treat it as a 'fixed object' and thus by "pulling against a fixed object...within reach...move as if [she] were climbing." The as if climbing tells us to use the climbing rules, which say that "Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling."
Thus, provided the rope is fixed, the warlock can move at half her movement by pulling herself closer to the sorcerer. Since this will be her movement, she is free to also blast away using her Cast a Spell action during her turn.
Now, why might the rope not be considered fixed? Well, the sorcerer is not attached to the ground - if the warlock were able to pull hard enough, she could move the sorcerer rather than herself. Since real-world physics do not apply, it is no use trying to figure mass, momentum, friction, and such. The DM must decide whether the sorcerer is big enough to count as 'fixed', at least for the warlock pulling on the rope (it may be helpful to remember that for the purposes of the rules, 'fixed' and 'immovable' max out at 8000lbs of weight regardless of what is attempting to move them).
The sorcerer can also move the warlock by carrying her
Normally, when a creature attempts to move another creature or object, they are interacting with it directly. However, the rope presents a complication, in that there are not specific rules for moving a creature or object through the use of an object. It seems reasonable for the DM to allow the sorcerer to move the warlock through the use of the rope (after all, mounts can pull wagons through the use of a harness), but one should be aware that this is a ruling, not RAW.
If the sorcerer is permitted to move the warlock via the rope, then, the next question becomes whether the warlock is being grappled, dragged, or carried? This has important implications for the movement speed of the sorcerer, for if they are in fact grappling the warlock, their own movement is then at half speed as per the grappling rules. If, however, they may be permitted to drag or carry the warlock with the rope, their movement might not be so impeded.
Normally, a creature is considered grappled, while an object is dragged or carried. However, the grappling rules assume that the target of the grapple is both resisting the grapple itself and resisting the attempt to move it. When this is not true, as here when the warlock is in fact willing to be moved by the sorcerer, it appears to be RAI to allow the mover to treat the person moved as an object; that is, to drag or carry her.
The next question is then whether the warlock is being dragged or being carried. As other answers to this question have said, nothing in the Levitate spell makes the warlock or her gear weightless, so all of this weight must then be considered. If the addition of the warlock and all her gear to the equipment already being carried by the sorcerer is in total still less than Sorcerer's 'carrying capacity'(their Str x 15, in pounds), then the sorcerer can still move at their full movement speed! Thus a strong sorcerer and a light Warlock can take maximum advantage of the levitating warlock by moving her rapidly around the battlefield. [Note: If the optional encumbrance rules are being used, full speed for the sorcerer is obtainable only when the total weight carried is less than (5 x their Strength score) rather than (15x).]
If, however, the total load of the sorcerer's gear plus the warlock and her gear exceeds 15 x the sorcerer's Strength (up to a maximum of 30 x Str), then the sorcerer cannot carry the warlock, and instead must drag her. In this case, the speed of the sorcerer is only 5 feet per turn, which will make this trick practically useless.
Conclusion: Given a strong sorcerer, a light warlock, and a willing DM, you can indeed run your warlock all over the battlefield on the sorcerer's turn.
Further considerations: the dangers of a slack line
A DM trying to add verisimilitude would do well to treat the rope as non-elastic, and to allow the sorcerer to move the warlock only when the rope is taut. Thus if on the sorcerer's last turn they moved the warlock north, are currently 40 feet north of her, and want to begin to move her south, they must first run 80 feet south before they can even begin to move her!
On both the sorcerer's and warlock's turns, they should be taking up slack. Since taking up slack in the rope is manipulating the rope but not moving either one of them, it would be treated as a free object interaction. The DM would need to determine how much rope can be moved this way on one turn before the Use an Object action became necessary (considering that this action would prevent the sorcerer from Dashing and the Warlock from attacking).
A slack rope presents other dangers, however. If the rope were to go so slack that part of it touched the ground, the DM might rule that it is no longer being carried - and thus would burst into flame! This is because if the rope is not carried, then the sorcerer's flame stride will burn through it. "When you move within 5 feet of a creature or object that isn’t being worn or carried, it takes 1d6 fire damage from your trail of heat." Since the rope, whether hempen or silk, has only 2hp, it would most likely be burned through on the first turn of the sorcerer in which it was not carried, stranding the suspended warlock.