The EX Treasure: Fairy Clothes mission will point you in the same direction as the Phantom Armour mission, as well as the Midna and Majora missions. Misko's EX journal drops the following hints for Tingle's outfit:
"The green garments of a man who wished to be a fairy are hidden in the ruins of Hyrule Field. Find these pieces are the ruins where trade flourished... at the ruins where sinners were imprisoned... and at a village ruins next to farmland where many people once gathered."
Where trade once flourished? Sounds like the Exchange Ruins. Look for this area to the northwest of Lake Kolomo, just off the road.
There are no enemies around here (unlike the other two locations you're about to visit), so it's a safe spot, albeit a fairly large one to explore.
If you've already got your Korok Mask, equip it here, because the chest is right near the Korok (if you've already found it, he'll show up as a seed on your map screen.)
Search the debris here and pull out the chest to get Tingle's Hood! As with all costumes and armor pieces in the DLC, if you sell this it cannot be replaced.
Here's the exact location on the map if you're struggling to find it.
Next, we're heading to a much more dangerous location - the Castle Town Prison, where sinners were kept. It's right next to the castle, so there's no quick and easy way in - I tend to fly down from Ridgeland Tower, but you're still liable to get attacked by horse-riding bokoblins on the way there.
Once you're there, there's a Guardian patrolling the area. Fighting this thing is all but unavoidable if you want the treasure, so make sure you equip a one handed weapon and a shield (if you've got the Master Sword, that's your best bet).
Deflect laser attacks with your shield (tap 'A' has as they're about to hit) and slash the legs off with your sword to take the Guardian down. Now you're free to look around.
Enter the prison ruins. There's not much to see here, although if you happen to have a rusted shield there's a Korok seed to be nabbed in here by placing it in a basket between two other rusted shields.
The chest is right near what looks like the "entrance." Use magnesis to find it.
Here's the exact map point I'm standing at as I open the chest.
Tingle's shirt! Kooloo-Limpah!
The last location, holding Tingle's tights, is the most obscure. It's called Mabe Village Ruins, and it's southeast of the castle, just off the main road. On the map above, I am standing in the ruins.
Head in there from any direction, but watch out for Guardians. Luckily the field around it is wide enough that you can safely keep your distance. Riding in on horseback (which, evidently, I did not do) is probably a good idea.
The ruins hold a few treasure chests - there's a silver rupee in the building in front of me in this shot - but otherwise it's a fairly innocuous location.
If a Guardian spots you, it's easy enough to hide until it goes away. We're looking for a Guardian, actually, albeit one that has already been destroyed.
Search the ruins and this Guardian should stand out. The chest is under it.
Again, map for reference - this is where you want to be.
Circle the Guardian with magnesis until you find the chest, then lift it out.
Tingle's outfit is complete! You can now dress as Tingle as your heart desires. Tingle, who was first introduced in The Legend of Zelda: Majora's Mask, is a bit of an eccentric, and it's nice to see him finally represented, in some small way, in BOTW.
TIngle's outfit doesn't grant much armor strength, but it lets you run faster at night, which is handy if you're being attacked by skeletons.