There are no rules for moving restrained creatures
There is nothing in the rules describing any kind of forced movement for restrained creatures. This means that whatever ruling will be a DM's call. However, I think there is a reasonable and straightforward ruling that is consistent with existing rules and easy to apply.
Adapting the rules for moving a grappled opponent
The closest we have in the rules is the rule for moving a grappled opponent. Luckily for us, entangled and grappled are very similar conditions and thus it makes sense for us to adapt the grapple rule for application here.
When you move, you can carry or drag the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Proposed house rule
This is the rule I would propose:
When your movement would also move an opponent entangled by a Rope of Entanglement, make a Strength (Athletics) contested by the target's Strength (Athletics). If the creature is two or more sizes smaller than you you automatically succeed on this check. If the creature is larger than you by one size or more, this check automatically fails. On a success, you can drag the entangled creature with you, but your speed is halved. On a failure, you cannot move with the entangled creature this turn. You can move normally if you drop the rope or release the creature from it.
This is very similar to moving a grappled opponent which is intentional and logical. However, there is an additional strength check involved which is there because, in a grapple the creature has already won in a contest and is using that to drag the opponent. In this case, however, a magic item has made that equivalent check to entangle the opponent and the item has no ability to drag something on its own. Thus, it seems reasonable to require an opposed strength check to simulate the act of pulling the restrained creature. It also happens to counterbalance the fact that the rope can do restrain someone from 20 feet away, avery powerful ability.
The fact that restrained and grappled are so similar make it very easy to see how we can adapt one rule and apply it to this. And the ruling comes out fairly easy to understand and apply in general.
The other options:
Would the enemy be automatically freed from the rope? - Probably not
The rope lists the conditions for a creature being freed from the rope and none of them include trying to move the creature. The only way this option would be feasible would be if the DM ruled that the rope would be good at entangling someone, but would otherwise not secure them enough to allow them to be dragged by it. If that was intended however, that kind of restriction would have been written into the item description. Also, it is a magic rope; tying things and dragging things are what rope does.
Would the rope take damage or break? - No, much tougher than normal rope
Though no necessary for the above rule I thought it might help to answer this a bit as well. The Rope of Entanglement is actually more durable than a normal hempen rope. Normal hempen rope has 2 HP and AC 11 whereas the Rope of Entanglement has 20 HP and AC 20 plus it has health regeneration. So it is significantly stronger and more durable than normal rope. As such, I do not think that this option is going to be a concern at all.