No, you will not gain infinite turns with this combo.
To break down the interaction:
- You control a Wormfang Manta (and have already skipped one turn).
- You cast Animate Dead targeting your Worldgorger Dragon, animate dead resolves and your Worldgrorger's ETB triggers.
- Worldgorger's ETB exiles all your permanents, including Animate Dead and Wormfang Manta. Wormfang's and Animate Dead's leave the battlefield effects trigger (you can choose the order they go on the stack, I will choose Wormfang on top of Animate Dead).
- Wormfang's LTB resolves, you gain an extra turn after this one. You now have +1 turn after this one.
- Animate Dead's LTB resolves, you sacrifice Worldgorger Dragon. Worldgorger's LTB triggers.
- Wormfang Manta and Animate Dead enter the battlefield (you choose the order they go on the stack, I will choose Wormfang on top of Animate Dead). You target Worldgorger with Animate Dead.
- Wormfang Manta's ETB resolves you skip your next turn. The next time you were to take a turn, you will now skip it instead.
- Animate Dead resolves, you bring Worldgorger back and start the whole thing over again.
So you can continuously flicker the Worldgorger with animate dead. Each time Wormfang Manta leaves the battlefield you will get an extra turn, but each time it enters the battlefield you will need to skip your next turn. So effectively you are back where you started again.
If you are looking for some infinite turns with Wormfang Manta, perhaps try something with Torpor Orb and something to flicker your Wormfang like Eldrazi Displacer?
The crux of the matter is that Wormfang Manta leaves the battlefield, giving you an extra turn, but is then returned to the battlefield, causing you to skip your next turn. Since your next turn is the extra one, it is skipped. Effectively, the two effects cancel one another.
You get a bunch of extra turns, but unfortunately you have to skip all of them.
Let's walk through the steps to get a clear picture of what happens. I'll refer to "enters the battlefield" and "leaves the battlefield" triggers as ETB and LTB, respectively.
- Animate Dead resolves.
- Animate Dead's ETB trigger triggers.
- Animate Dead's ETB trigger resolves. You return Worldgorger Dragon to the battlefield and attach Animate Dead to it.
- Worldgorger's ETB trigger triggers.
- Worldgorger's ETB trigger resolves. You exile Animate Dead and Wormfang.
- Wormfang's LTB trigger and the delayed trigger set up by Animate Dead both trigger. It doesn't matter how you order the triggers. Resolving Animate Dead first would result in a large number of Wormfang triggers all resolving at the end of the loop, and the route we'll take will have them resolve one at a time.
- Wormfang's trigger resolves. You get an extra turn after this one.
- Animate Dead's delayed trigger resolves. You sacrifice Worldgorger.
- Worldgorger triggers.
- Worldgorger's trigger resolves. You return Animate Dead and Wormfang to the battlefield.
- Wormfang's ETB and Animate Dead's ETB triggers both trigger. Again, it doesn't matter how you order the triggers. If we wanted to, we could build up a large pile of Wormfang triggers and resolve them all at the end of the loop. Instead we'll choose to resolve Wormfang's trigger first.
- Wormfang's ETB trigger resolves. You skip your next turn.
- Return to step 3.
Next, we need to establish that "skip your next turn" is a replacement effect.
614.1b Effects that use the word “skip” are replacement effects. These replacement effects use the word “skip” to indicate what events, steps, phases, or turns will be replaced with nothing.
And finally, we need to show what happens if you have a hundred replacement effects waiting to replace the same "next" event.
616.1. If two or more replacement and/or prevention effects are attempting to modify the way an event affects an object or player, the affected object's controller (or its owner if it has no controller) or the affected player chooses one to apply, following the steps listed below.
That means replacement effects apply one at a time. Therefore, each time you would take an extra turn, instead one replacement effect applies and you skip the turn. The other replacement effects no longer apply to that particular event, so they continue waiting to replace the next turn.