Yes, it does. The way to enable this depends on your version of Notepad++. On newer versions you can use:

Menu View → Show Symbol → *Show All Characters`

or

Menu View → Show Symbol → Show White Space and TAB

(Thanks to bers' comment and bkaid's answers below for these updated locations.)

On older versions you can look for:

Menu View → Show all characters

or

Menu View → Show White Space and TAB

The simplest way is to use the menu:

View->Show Symbol->Show all characters

It will show Tabs and EOL characters.

Here is an example of a VBS script that announces the current hour being viewed in Notepad++ on Win7.

Note that it is clear I used spaces to do the indents - not Tabs.

This is the base script I used in the Q&A about how to make Windows announce the time at the top of each hour:

https://superuser.com/questions/1268702/how-to-make-windows-speak-the-time-at-the-top-of-the-hour/1268703#1268703

Notepad++ allows you to customize your shorcuts. You simply need to go to the Settings menu and then go to the Shortcut Mapper option.

You can check out this tutorial.

BBEdit is a really great Mac text editor, and can show invisible characters. http://www.barebones.com

In addition to DrZoo's answer, there is a second possibility of where the error comes from: Your curent font does not have a glyph for the fox. Especially since Emojis are new-ish and low on the priority of font-makers, it is quite likely that your font simply cannot provide a symbol, in which case notepad++ defaults to the square you see.

Check what font you have and then verify that it indeed has emojis, or switch to a font known for having them. One of these would be "consolas", which is used here on stackexchange (and given that you do have the fox in your post, we know that it includes the symbol you need)