I've had constant trouble with this since I built my new computer just over a month ago, and have done a decent amount of digging, researching, and experimenting to better narrow down the issues. Figured I'd make a post on this to better collect information, and at least have a decent description of the problem for anyone digging around themselves.
The symptoms: The DirectX 11 client of FF14 crashes randomly while playing on AMD cards using the Tonga chipset, this includes the R9 300 series as well as the R9 285(among others). No specific in-game event or instance seems to be causing these crashes, it has happened to numerous people everywhere in the game at any given time. It can be 10 seconds, 10 minutes, or 10 hours into playing, but the game WILL crash. Giving the Error "A fatal DirectX Error has occurred(11000002)", crashing out, or sometimes locking up the computer entirely (Though I've only experienced the first of these). It is also important to note these issues do not happen in the DX9 client, and do not happen in other DX11 games. This seems to a problem unique to FF14 and this range of AMD cards (though I have heard Witcher 3 having the exact same issues, I don't own the game so can't comment too well on that).
The Issue: The biggest issue with narrowing down anything specific is that this problem does not seem tied directly to the game itself. Changing settings doesn't appear to affect anything (and is hard to experiment with, given the randomness of the crashes themselves), either in the game or in Catalyst Control Center for the GPU itself. After a couple weeks I'm confident in saying the main problem is one of two things: The FF14 DX11 client does not properly work with AMDs Tonga drivers -or- This specific chipset of AMD cards really, REALLY doesn't like FF14s DX11 client. I'm really not sure which side of the fence this problem falls on (either AMD or Squeenix), but in the end it's still a problem. Best-case scenario is that it's a driver issue AMD hasn't solved yet or a client issue Squeenix hasn't solved yet. Worst-case scenario is that the DX11 client just can't run on Tonga cards, and we're doomed(well, "doomed") to run the game in DX9 forevermore.
Workarounds: The best workaround, as stated right above, is to simply play the game in DX9 mode. I don't consider this a solution in the least, but the game is simply unplayable in DX11 mode when you can crash at literally any second. I've heard some success of messing around with some settings, but the closest thing to "success" is playing the game for slightly longer than normal before it crashes. And given how random these crashes are I'm hesitant to honestly call any of these a genuine success when it could just be luck. The safest bet is to stay in DX9 until something is actually fixed, or at the very least acknowledged. As far as I know, neither AMD nor Squeenix have commented on this issue which further confounds things a bit. At least hearing from one or the other on would go a long way towards fixing this. Here's hoping...