Why facebook videos are not playing in my smartphone?

There must be some issue with your internet browser. You can follow these steps which may help you to solve your issue.
1. Update your Internet Browser - If there is update available for your browser then you should install it because Facebook have recently updated few parts of their website. Apart from that clear your browser's cookie and cache because they save some parts of the website for faster access. So clear them out to get a fresh files from data server.

2. Download Facebook App - I am pretty amazed that you are choosing to go for old style web browser Facebook whereas there is a pretty awesome application waiting for you to just download it. I will prefer you to download latest updated facebook application because it will be able to play all its assets.

3. Download 3rd Party Browser - You can download UC Browser from Windows App Store which will let your play videos online.

If you are still facing issue then I will recommend you to update your IS and perform a hard reset.

If the Internet connectivity on your device is on the slower side, you may face issues in playing the video files.
The default browser sometimes does not play the videos. Install a third party browser and see it improves the situation. Chrome is the best browser both on your phone and desktop. If you have the issues of slow Internet connection, switch to UC Browser which is the best one when it comes to faster video playing abilities.
To check if your Internet connection is the culprit, you may connect to the service using WiFi.
Instead of using a browser, use the official Facebook app and check if it can play the videos. The Web version of Facebook can have issues at times.
If all this fails, you may need to reset your phone to rule out the problems with it if any.

Nokia Lumia has Internet explorer for mobile version 11 installed on it. So the thing is that it yet to be fully compatible with the HTML5 video streaming. There are going to be some glitches for the same. You can install either Opera Mini or other browser that supports HTML5 videos. Facebook currently only has one flash version and one HTML5. If for your phone they are streaming the flash version or HTML5 then you may face issues.

Three things could help you with this -

1. Change the browser.
2. Make sure you use Facebook native app for Windows mobile.
3. Install the necessary plugin for the mobile browser that solves the streaming issue.

I think changing the browser or installing the facebook native app should solve this issue. In worst case scenario, you can ask Microsoft support for resolution of this issue.

The answer could be more specific, if you let us know, whether it is playing other videos from other sites.
If yes, then it has to be the issue of video format support. If no, then there can be few more reasons like:

1) Memory issue, you need to free some memory.
2) Check your internet pack, if it has exceeded the data
3) Might be some virus catching up your device

So, to ensure all the above,
a) check your phone's memory status
b) check your internet data pack
c) you need to load and run a good antivirus.

Hope that works.