So I've been looking at the path of the beast for a while now, and suddenly I had an idea.  What if when I raged I decided to grow the tail, an I was holding a greataxe.  Could I attack with the tail as an action, then use the two weapon fighting bonus action to attack with the greataxe?  And if so, since it wasn't in my off hand, would I get my modifier to the damage roll?

It has to be a light weapon 

What if I take the dual wielder feat?

No, since it is a natural weapon. It is not a "one-handed weapon" nor are you wielding it.

Unlike the Loxodon's Trunk, the Path of the Beast's Tail says nothing about being able to grasp, so it seems that RAW it cannot, which means it can only be used for natural weapon attacks, not wielding weapon attacks.

You could ask your DM for permission to let it grasp however, as nothing says it cannot (grey area).

So RAW I would say natural weapon attacks only, but DM's choice (if not in Adventure League).

The RAW is very unclear whether "Natural Weapons" are intended to be a fourth type of Weapon (Simple Weapons, Martial Weapons, Improvised Weapons), or instead are a non-weapon Unarmed Strike. Depending on where you look in the PHB or MM, you'll find language to support your preferred interpretation. But even if they are, the answer is still probably no.

Two Weapon Fighting is a rather complicated decision tree (which Dndbeyond forum formatting is not going to help make any easier)

Two-Weapon Fighting

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

1. To qualify to make a Bonus Action TWF attack, you must:
   1. Be holding in Hand1:
      1. Weapon 1
      2. That is Light
   2. Be holding in Hand2 a:
      1. Weapon 2
      2. That is Light
   3. Make an attack
      1. Using the Attack action
      2. With Light W1 or W2 in H1 or H2 (doesn't matter which one is your "main" or "offhand" weapon)
2. This enables you to:
   1. make an attack:
      1. with a Bonus Action
      2. with a Weapon
         1. that is Light
         2. that is a different weapon from the triggering attack
         3. that is held in a different hand from the triggering attack

Dual Wielder relaxes some of the above, but not all of it. All it really does is strike the "is Light" conditions from all of the above, not anything else.

Is a Natural Weapon a Weapon? That's a necessary, but not sufficient, part of being able to use it for Two Weapon Fighting. Your mileage may vary with your DM.

Is a Natural Weapon held in a hand? That's a necessary, but not sufficient, part of being able to use it for Two Weapon Fighting. Very unlikely that a Tail attack  (is your tail a third "hand"? Maybe, if its graspy, but probably not...) or a Horn or Hoof or Bite attack would qualify. Maybe a Claw would? Again, your mileage may vary with your DM. 

There's several rule dominos that need to line up for your DM to buy in to Natural Weapons being usable in Two Weapon Fighting. My own agreed to it the other night for a one shot, but that's probably only because I got him drunk first. :p

Quote from Chicken_Champ >>

The RAW is very unclear whether "Natural Weapons" are intended to be a fourth type of Weapon (Simple Weapons, Martial Weapons, Improvised Weapons), or instead are a non-weapon Unarmed Strike. Depending on where you look in the PHB or MM, you'll find language to support your preferred interpretation.

 Including an unarmed strike being a weapon. See Githzerai Monk with "Unarmed Strike. Melee Weapon Attack: "

Quote from Pantagruel666 >>

Quote from Chicken_Champ >>

The RAW is very unclear whether "Natural Weapons" are intended to be a fourth type of Weapon (Simple Weapons, Martial Weapons, Improvised Weapons), or instead are a non-weapon Unarmed Strike. Depending on where you look in the PHB or MM, you'll find language to support your preferred interpretation.

 Including an unarmed strike being a weapon. See Githzerai Monk with "Unarmed Strike. Melee Weapon Attack: "

 Unarmed Strikes use Melee Weapon Attacks, but are not “Weapons,” whereas “Natural Weapons” are Unarmed Strikes that do count as weapons. It was always intended to be that way, and many people have always interpreted that as RAW. In addition, the latest Sage Advice Compendium clarified that as RAI too.

Fortunately the RAI is pretty darn obvious that you should not be using TWF with a heavy greataxe and a natural weapon that is the equivalent of a halberd. It's fun to bring it up in a rules forum, but at most tables you would get laughed out of the room just for bringing it up. I mean I'm all about saying yes to my players but they gotta be reasonable.

Quote from Spectralite_Quago >>

So I've been looking at the path of the beast for a while now, and suddenly I had an idea.  What if when I raged I decided to grow the tail, an I was holding a greataxe.  Could I attack with the tail as an action, then use the two weapon fighting bonus action to attack with the greataxe?  And if so, since it wasn't in my off hand, would I get my modifier to the damage roll?

The answer is: absolutely not, under no circumstances is that permitted by RAW (and/or RAI)

• A Greataxe is a Two-Handed weapon; absolute disqualification right here, full stop. No way around this.
• Natural weapons are considered actual weapons, now, for rules purposes. They still do not ever qualify for use with TWF.
  • Natural weapons do not have the Light property. The Dual Wielder feat negates this requirement.
  • Natural weapons are incapable of being held. The Dual Wielder feat does not negate this requirement.

The easiest thing is to mechanically play RAW but flavor-wise describe it as wielding an axe with your tail.

Mechanically playing RAW does not ever allow the use of Two-Handed or natural weapons with TWF, no matter how you describe it.

You could use a two-handed weapon as part of TWF if you had Dual Wielder and at least three hands and traits that allow you to use them all (might be an issue if they publish a playable Thri-Kreen race), but RAW if the intent is to allow using a natural weapon as a secondary attack there will be a trait specifically allowing doing so.

Quote from Pantagruel666 >>

You could use a two-handed weapon as part of TWF if you had Dual Wielder and at least three hands and traits that allow you to use them all

Uhh... no, they absolutely cannot ever do that.

Two-Weapon Fighting

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

Dual Wielder

You master fighting with two weapons, gaining the following benefits:

• You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.
• You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren't light.
• You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

There are absolutely zero circumstances in which a weapon with the Two-Handed property is eligible for use with Two-Weapon Fighting.

There are absolutely zero circumstances in which a Natural Weapon is eligible for use with Two-Weapon Fighting.

If you somehow had three hands and Dual Wielder, and were holding a Versatile one-handed melee weapon in two hands and a one-handed in the other, then Dual Wielder language wouldn't be a problem... but "When you take the Attack action and attack with light melee weapon that you're holding in one hand" from the basic TWF entry still would be. Sigred has a point.

There isn't anything that explicitly says that holding something in two hands fails to generally satisfy holding in one hand ("hold in one or more hands," essentially), but the plainest reading of "one hand" is "not two hands," especially in this context, so you'd really have a RAI hill to get over with that. 

I mean, really, there is no RAI hill to get over. It's a full-on "no, you just can't do that".

"One-Handed" is not a weapon property; it's a context. A Versatile weapon being wielded with two hands is not "One-Handed". A weapon is "One-Handed" when actually wielded in one hand.

The fact that Two-Handed is a weapon property, and that weapons are referred to in some places as "one-handed weapons" instead of just as "weapons wielded in one hand," strongly implies that One-Handed is an unwritten-but-implied property of all weapons that are not Two-Handed. But, unwritten, so it's a judgment call. 

Quote from Chicken_Champ >>

The fact that Two-Handed is a weapon property, and that weapons are referred to in some places as "one-handed weapons" instead of just as "weapons wielded in one hand," strongly implies that One-Handed is an unwritten-but-implied property of all weapons that are not Two-Handed. But, unwritten, so it's a judgment call. 

I hear what you're saying. I even agree, more-or-less, but I don't think it's consistent for answering the question at hand. [edit]... I S2G I didn't mean to make that pun...

"One-Handed" is a context, not a property, implied or otherwise. I do agree that it is also the default context for how a weapon weapon is wielded. That context, in no uncertain terms, fundamentally cannot be inclusive of a weapon being wielded in any other manner.

If a "one-handed melee weapon" meant anything other than literally "a melee weapon being wielded with one hand", "One-Handed" would be an explicitly stated property.

If there were even the slightest possibility of Two-Handed (as in the context of it being wielded with two hands) weapons, or Versatile weapons being wielded with two hands, being eligible for use with TWF, they would have said as such.

TWF does not care about whether your weapons have Two-Handed or Versatile properties; it cares about whether or not your weapons are actually being used with one hand each. If something like Monkey Grip (please, no) were to make it into 5e--allowing the creature to use Two-Handed weapons in one hand, or a Versatile weapon's two-handed damage dice with one hand--that would permit the use of TWF with the correct context.

Hopefully, if something like that got introduced, it would be explicit in the trait description. You'd actually have to specify quite a bit to make a playable Thri-Kreen because anything referring to the 'other' hand doesn't make sense.

Quote from Pantagruel666 >>

Hopefully, if something like that got introduced, it would be explicit in the trait description. You'd actually have to specify quite a bit to make a playable Thri-Kreen because anything referring to the 'other' hand doesn't make sense.

"Other hand" is still just context, not property-dependent. It doesn't matter how many hands you have if the weapon(s) you're using require more than one hand to wield, or are being wielded by more than one hand. If you've got 3+ hands to work with, TWF only cares about the ones you're actually using with it. You could have two hands each wielding an applicable, independent weapon, and two more hands holding a Greataxe, you would still be able use TWF with the two one-handed weapons; you would not be able to use the Greataxe for any of those attacks.

[edit]But yeah, Thri-Kreen would/should need a lot of technical specificity to be playable.