Hi! Here is solution for Opera 43. (Kubuntu 16.04 x64)

1. We need to update Chromium browser to 56.0.2924.76 version (Opera 43 use this version of Chrome engine)
2. Add repository - sudo add-apt-repository ppa:canonical-chromium-builds/stage
3. Update system - sudo apt update
4. Chromium must updated to 56 version
5. As recently as advised tyrion-ua:

Copy the libffmpeg.so file from the library (/usr/lib/chromium-browser)
Replace the libffmpeg.so file in the browser folder (/usr/lib/x86_64-linux-gnu/opera) with this file.
Restart the browser.

Now Opera plays videos from Twitter, Instagram, etc.

Yes! It really works. Thanks avl.
mrmatteastwood, find the package you can find using the search line on launchpad.
Like this

and this page

Then you select the appropriate version for your system, download and install package.

Bookmark this page https://launchpad.net/ubuntu/xenial/+package/chromium-codecs-ffmpeg-extra
and check the version of the engines chrome and opera.
Sort of like working

Thank you in-tensive. After a little hokey pokey I got this to work. Thank you for helping me learn a little more about Linux and how to fix this very annoying bug.

Not at all, kpsaver.
I am a fan of Opera Browser and happy to help you.
Despite some of the problems of our beloved web-browser ;)))))

Solution in this discussion -

https://forums.opera.com/topic/19182/opera-browser-does-not-play-twitter-videos-and-gifs-again/17

tyrion-ua,

It was your idea. I just modified it.

Hi! Here is solution for Opera 43. (Kubuntu 16.04 x64)

1. We need to update Chromium browser to 56.0.2924.76 version (Opera 43 use this version of Chrome engine)
2. Add repository - sudo add-apt-repository ppa:canonical-chromium-builds/stage
3. Update system - sudo apt update
4. Chromium must updated to 56 version
5. As recently as advised tyrion-ua:

Copy the libffmpeg.so file from the library (/usr/lib/chromium-browser)
Replace the libffmpeg.so file in the browser folder (/usr/lib/x86_64-linux-gnu/opera) with this file.
Restart the browser.

Now Opera plays videos from Twitter, Instagram, etc.

Open a termnal windows and type sudo apt-get install --reinstall pepperflashplugin-nonfree.

Just checked. It really works.