It works fine in the editor (3.5.3, Windows), but the exported version fails with the below text.
I'm importing .exr files as Image so I can get pixels as a depth map. They are half-float RGB from a Blender Depth pass, ZIP-compressed.
I've also tried full-float, no compression, etc. Same error results.
I tried importing then exporting from Krita in case there is a Blender format issue but I got the same result.

Loading resource: res://0100s/N0101_BarnDeckCenter/BG0101_n.jpg
Loading resource: res://0100s/N0101_BarnDeckCenter/DP0101.exr
ERROR: Method failed. Returning: RES()
   at: load (core/io/image_loader.cpp:162)
ERROR: Failed loading resource: res://.import/DP0101.exr-f83cd5e36cc01191cf565ee5e3f65cf5.image. Make sure resources have been imported by opening the project in the editor at least once.
   at: (core/io/resource_loader.cpp:270)
ERROR: Failed loading resource: res://0100s/N0101_BarnDeckCenter/DP0101.exr. Make sure resources have been imported by opening the project in the editor at least once.
   at: (core/io/resource_loader.cpp:270)
SCRIPT ERROR: Parse Error: Can't preload resource at path: res://0100s/N0101_BarnDeckCenter/DP0101.exr
          at: GDScript::load_byte_code (res://CommonScripts/g.gdc:318)

follow up:
I gave up and found a way to use 16-bit PNGs instead, which don't register the same problem. I'd rather use floats though than having to precisely scale my PNG values to cover the same large range as the EXRs will hold. (I'm working on a non-linear scaling of the data before it is saved as a PNG then reverse the non-linear operation to recover values with higher resolution when small, just like floats)

    15 days later

    Shadowphile
    well this sucks. Godot does not read 16-bit PNGS so I've actually been rendering with 8-bit images. If only EXR files didn't have the export problem!