There is a style issue you can fix. See https://docs.godotengine.org/en/stable/contributing/workflow/pr_workflow.html

Run:

pip3 install pre-commit
# Go to Godot Engine folder
pre-commit run -a

Not sure where the proper directions are for this.

I could provide a screenshot of it working, although there is not much to see. We tested this with the unit test and a scene where we load an EXR file and set it in a TextureRect for display.

Documentation changes required.

Try godot --doctool

diff --git a/doc/classes/Image.xml b/doc/classes/Image.xml
index 97c4bba..138abe0 100644
--- a/doc/classes/Image.xml
+++ b/doc/classes/Image.xml
@@ -346,13 +346,6 @@
 				[b]Note:[/b] This method is only available in engine builds with the BMP module enabled. By default, the BMP module is enabled, but it can be disabled at build-time using the [code]module_bmp_enabled=no[/code] SCons option.
 			</description>
 		</method>
-		<method name="load_from_file" qualifiers="static">
-			<return type="Image" />
-			<param index="0" name="path" type="String" />
-			<description>
-				Creates a new [Image] and loads data from the specified file.
-			</description>
-		</method>
 		<method name="load_exr_from_buffer">
 			<return type="int" enum="Error" />
 			<param index="0" name="buffer" type="PackedByteArray" />
@@ -361,6 +354,13 @@
 				Note: The TinyEXR module is disabled in non-editor builds, which means load_exr_from_buffer will return an error in exported builds unless it is explicitly enabled at build-time using [code]tinyexr_always=yes[/code] SCons option.
 			</description>
 		</method>
+		<method name="load_from_file" qualifiers="static">
+			<return type="Image" />
+			<param index="0" name="path" type="String" />
+			<description>
+				Creates a new [Image] and loads data from the specified file.
+			</description>
+		</method>
 		<method name="load_jpg_from_buffer">
 			<return type="int" enum="Error" />
 			<param index="0" name="buffer" type="PackedByteArray" />

Will those commits be squashed on merge?

No, once this is approved you'll need to squash your commits, but it's not necessary now

A tinyexr_always build option was added to enable TinyEXR even in template builds.

This is also implemented by #73003, although it uses the tinyexr_export_templates=yes SCons option instead (which I think is more explicit). Your implementation of the option seems cleaner still, so I'm in favor of this PR over #73003.

I think we should rename the option in this PR to tinyexr_export_templates=yes, or just make use of module_tinyexr_enabled=yes being explicitly defined like #94965 did for runtime lightmap baking.

What's the size impact on optimized export templates to enable tinyexr at runtime? We need to evaluate the cost to see if we have to bother adding a SCons option and disabling this by default (which means hardly anybody will use it).

The other PR (#73003) measured a size increase of 100KB for export templates. I assume that value did not change drasticly since then.

I encourage enabling tinyexr by default.

Should I commit changes to enable tinyexr by default within this PR or create another one?

It's been quite some time... I just rebased my changes to work with version 4.5.
What is left to do for the PR to be accepted? Is it only the missing review from AThousandShips?

The requested change was implemented. Is there something else missing for this to be merged?

Could you squash your commits? See our pull request guidelines for more information

Rebased and squashed.

Oops. Good you noticed that. Its fixed now.

Thanks! Congratulations on your first merged contribution! 🎉