Thought I might do things a bit differently this time. Let's start with this:
https://github.com/invertego/tex2png

And for people who can't/don't care to build from source:
Prebuilt Windows binaries (requires Vista/7 or later and .NET Framework 4.5)

What is it?
A command line tool for converting images as they are natively stored in the iOS version of Phoenix Wright: Dual Destinies - files with the ".tex" extension - to a more readily usable format (PNG).

How do I use it?
Well, first you need a copy of Phoenix Wright: Dual Destinies for iOS (so far I've only tried the English version, but Japanese should work too). If through some means - justified by the ends, no doubt - you can get the IPA file containing the game onto a PC, you can run it through this tool to convert the textures therein to PNGs. Or you can just download the whole pack here.

What will I get?
A bunch of random things, most notably full screen images like this:

What is missing?
Anything that isn't stored in a standalone TEX file. There are a number of game files with the extension ".arc", and these appear to be archives containing additional game data. On the plus side, much of the game's 2D art is in standalone files.

...and I think that's about it.

Thanks for doing this!
If only there was a way to open those .arc files. Still, things like the thought route art and the beta Apollo profile image are pretty neat!

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IIRC, Nintendo uses an archive format also called .arc. Maybe they're actually the same format, considering how DD iOS is a 3DS game port and all?

If that's the case, doesn't that mean it'll probably harder to crack them open? (Or have people worked it out with the 3DS, and we could apply the same principle here.)

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.arc was also used in Wii games

Looks like the .tex files on the iOS release are in a different format than that of the .tex files on the 3DS release...
Either that or it's just the .tex files I'm trying to convert...

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The .ARC files used in DD iOS are encrypted.
Some of the ones in the 3DS release are not.

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