Is it possible for two player characters to move as a group by taking turns grappling each other?
Assuming a 30' speed for each player (each with the Grappler feat), and no magic items, spells, or class abilities, how far could they get in a single round?
I assume they will each move and dash, and I think they're going to need to take advantage of opportunity attacks somehow to switch the grappling and grappled creature.
I'm assuming this would work via the fact that the Grappler feat doesn't slow your movement, characters can willingly fail saves, and that a character can release a grappled target whenever they want.
Yes, this works from a RAW perspective.
In total, they moved 60' (both had to use their Action to initiate the grapple, so it was unavailable to Dash).
While this kind of exploit doesn't meet the cheese threshold of the peasant railgun, you should be aware that the DMG does specifically advise against allowing this kind of loophole:
Rules Aren’t Physics. The rules of the game are meant to provide a fun game experience, not to describe the laws of physics in the worlds of D&D, let alone the real world. Don’t let players argue that a bucket brigade of ordinary people can accelerate a spear to light speed by all using the Ready action to pass the spear to the next person in line. The Ready action facilitates heroic action; it doesn’t define the physical limitations of what can happen in a 6-second combat round.
So unusual acceleration due to quirks in the rules should generally be disallowed.
and also:
Combat Is for Enemies. Some rules apply only during combat or while a character is acting in Initiative order. Don’t let players attack each other or helpless creatures to activate those rules.
In general, you shouldn't be using combat rules to engage in exploits with members of your party.
Rules Rely on Good-Faith Interpretation. The rules assume that everyone reading and interpreting the rules has the interests of the group’s fun at heart and is reading the rules in that light.
If it doesn't make logical sense logically, it's probably not the intent of the rules.
So your DM would be on firm ground to disallow this kind of exploit.
The DMG has this advice on p. 19 for players that aim to exploit combat mechanics in unusual ways by attacking their allies:
Combat Is for Enemies. Some rules apply only during combat or while a character is acting in Initiative order. Don't let players attack each other or helpless creatures to activate those rules.
Here, players are attacking each other to activate a rule that only works during combat, with the objective to gain a movement advantage. Don't let them, unless you prefer to ignore this guidance.
Rules-as-written, the characters could effectively double their normal movement at the expense of giving up their action. Each one could on their turn grapple the other using their action, then move at full speed (thanks to Grappler), then let go (no action cost), for a total movement of 60 feet. At baseline, this is no better than if they just used their action to Dash, so in that regard, they could not move further using this trick.
However, the game never plays without any class abilities, even if you excluded them in your question. With class abilities, this would shoot up - for example Rogues can Dash as a bonus action (or Monks can grapple with their Unarmed strike as a bonus action), increasing this to 120 feet. Exploits often come from combining multiple features. I do not think that this is a particularly strong one, as it has a pretty hefty cost in two Feats to work, so I don't think it needs to be banned from a balance perspective, but it also doesn't sit right from a verisimilitude one -- why would a couple of friends be faster to cover some ground by wrestling and dragging each other, than if they just ran as fast as they could?
You ask:
"Is it possible for two player characters to move as a group by taking turns grappling each other?"
There are already many great answers to this.
I would just like to add, you have the core mechanic of how to play D&D confused.
Your question supposes that the rules are in charge. They aren't. The players are in charge, specifically the DM. The idea that you can look at some rules and sort of squint at them sideways, you can then come up something absurd that a DM has to accept because "that's what the rules say" is just not true.
From "The Rhythm of Play", in the Free Rules:
In #2, in your scenario, a player says something like, "Alice is going to grapple Bob and (blah grapple blah blah dash, some rule exploit)".
Most DMs are going to respond with, "What are you on about? You're not in combat. What are you actually trying to accomplish?"
And once it becomes clear that on Bob's turn, Bob's going to grapple (or pick up or whatever) Alice, the DM is going to say something like, "Ah, right. Yeah, no, of course that isn't going to work. Remember, even in combat, we use turns to resolve things that are happening simultaneously. You're just exploiting that. It's a clever trick, sort of, but not a new one. Look up 'peasant railgun'. This is a great opportunity for us to discuss how we handle possible rule exploits like this, so that we don't bog down the game and we can all have fun."
So, actually, since the DM narrates the results of the adventurers' actions, a DM could say, "sure, I'll allow it", even though isn't not actually allowed by the rules, as pointed out in other answers. Most DMs won't allow it, though, because they'll judge it leads to a ridiculous outcome that makes the game less fun.
