Emoji do not appear anymore at all, a blank rectangle is displayed instead.
See https://github.com/brave/brave-browser/issues/7397#issuecomment-572756439
Open any website with emoji such as Twitter. Emoji can also be copypasted from Emojipedia. I had one in a bookmark which is also not displaying correctly anymore.

orange heart emoji
always
Brave 1.1.21 Chromium: 79.0.3945.79 (Official Build) (64-bit)
Revision 29f75ce3f42b007bd80361b0dfcfee3a13ff90b8-refs/branch-heads/3945@{#916}
OS Linux
JavaScript V8 7.9.317.31
Flash (Disabled)
User Agent Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/79.0.3945.79 Safari/537.36
Command Line /opt/brave.com/brave/brave --enable-dom-distiller --disable-domain-reliability --no-pings --extension-content-verification=enforce_strict --extensions-install-verification=enforce --enable-oop-rasterization=Enabled --sync-url=https://no-thanks.invalid --enable-features=PasswordImport,WebUIDarkMode,SimplifyHttpsIndicator --disable-features=AutofillServerCommunication,NotificationTriggers,AudioServiceOutOfProcess,SmsReceiver,LookalikeUrlNavigationSuggestionsUI,WebXR,WebXrGamepadModule,UnifiedConsent,AllowPopupsDuringPageUnload,SyncUSSBookmarks --flag-switches-begin --flag-switches-end --disable-webrtc-apm-in-audio-service
Executable Path /opt/brave.com/brave/brave
Profile Path /home/mitsunee/.config/BraveSoftware/Brave-Browser/Default
I don't know if graphics drivers are related to fonts in any way but I recently updated to nvidia driver 430, tho I have noticed a broken emoji in bookmarks about 2 brave updates prior.
You need to install fonts that support special characters like emoji. The browser will pick up the system font to show it. You might want to install fonts-symbola for your distro using the package manager or use Ubuntu package. This should support most if not all types of font emoji.
Going to close this out - but @Mitsunee please let us know if @srirambv 's suggestion fixes the issue 馃槃 If not, we can re-open and investigate more
This issue is not caused by a specific font-family. Emojis stopped working with a recent update in Brave only. Other software can display emoji just fine, even bash and gedit can.

The same emoji pasted into brave does not display anywhere, I tried this very input field I'm typing into right now, the devtools and bookmarks.

Please stop closing unsolved issues before even giving me a chance to respond.
It's re-opened (and you have ability to reopen)
for future reference, I upgraded to Kubuntu19.10:
Brave 1.1.22 Chromium: 79.0.3945.79 (Official Build) (64-bit)
Revision 29f75ce3f42b007bd80361b0dfcfee3a13ff90b8-refs/branch-heads/3945@{#916}
OS Linux
JavaScript V8 7.9.317.31
Flash (Disabled)
User Agent Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/79.0.3945.79 Safari/537.36
Command Line /opt/brave.com/brave/brave --enable-dom-distiller --disable-domain-reliability --no-pings --extension-content-verification=enforce_strict --extensions-install-verification=enforce --enable-oop-rasterization=Enabled --sync-url=https://no-thanks.invalid --enable-features=PasswordImport,WebUIDarkMode,SimplifyHttpsIndicator --disable-features=AutofillServerCommunication,SyncUSSBookmarks,AllowPopupsDuringPageUnload,WebXR,AudioServiceOutOfProcess,SmsReceiver,LookalikeUrlNavigationSuggestionsUI,NotificationTriggers,WebXrGamepadModule,UnifiedConsent --flag-switches-begin --flag-switches-end --disable-webrtc-apm-in-audio-service
Executable Path /opt/brave.com/brave/brave
Profile Path /home/mitsunee/.config/BraveSoftware/Brave-Browser/Default
Issue still unchanged, no emoji display in any textboxes, bookmarks or anything while they do display in most other software (spotify doesn't, but that's the smallest issue that software has on linux...)
+1 in Brave Browser 1.1.23 and Ubuntu 19.04
Previous versions of Brave Browser had emojis working, since last two updates (updates 1.1.x) stopped working.

+1 on Ubuntu 19.10 and Brave 1.1.23

Whereas in Firefox:

I had the same issue with the latest release (on Ubuntu 19.10) and fixed it by removing and reinstalling fonts-noto-color-emoji. Might help you people.
I realised this problem also occurs on Google Chrome, so is something realted to Chromium.
The solution from @chrsengel seems to work. I tried
sudo apt-get reinstall fonts-noto-color-emoji
and after closing all Brave / Google Chrome Windows and reopening is working in both browsers
@Mitsunee can you try the above suggestion please?
I already put a like reaction to show that I tried it and it worked. As I don't know how it worked I didn't leave a comment.
update: I keep having to repeat the workaround after every new update and/or reboot, which is especially annoying because I have to reboot a lot due to a problem with the nvidia drivers...
Most helpful comment
I had the same issue with the latest release (on Ubuntu 19.10) and fixed it by removing and reinstalling
fonts-noto-color-emoji. Might help you people.