Use the Interpolation plugin. It is not turned on by default so click Plugins -> Manage Plugins and check the Interpolation Plugin box. Once activated access the plugin by clicking Raster -> Interpolation -> Interpolation.
From the plugin's menu there is a checkbox that will allow you to use the Z value of your layer as the elevation source rather than a value from the attribute table.
If you have access to the 3D Analyst extension, this can be accomplished using the TIN Node tool found here: 3D Analyst Tools - Conversion - From TIN - TIN Node. This will create 3D points (i.e Z-enabled) at each triangle node that forms the TIN. Alternatively you could convert your TIN to a raster, and then use the Raster to Multipoint tool which will create points at each raster cell center (as opposed to the TIN nodes). This approach will give you more flexibility regarding the number of output points, as it will be based on the raster resolution which you can modify to fit your needs.
Once you have your point file, you can use the Feature Class Z to ASCII to get your desired output format.
Best Answer
It is an new feature in the future release QGIS 3.10. QGIS can now read ESRI TIN as this format is now supported by the MDAL library.
Edit :
Finally, it seems it will available in the future release QGIS 3.12 (end of February 2020).