As of version 1.0.0 NodeBB has by default enabled sendmail transport plugin. So it's enough to make available "sendmail" command in your system.
For example on Debian Jessie with postfix installed you can do so with symlink:
ln -s /usr/sbin/sendmail /usr/local/bin/sendmail
and test it like:
echo "Test mail" | sendmail -i -f email@example.com firstname.lastname@example.org
Ok, the problem was solved. But I'm not sure how exactly.
After I wrote this post I noticed that the latest version of NodeBB is 1.16.x. But while upgrading the forum i switched to 1.15.x branch (I thought that was the latest).
After that, I used ./nodebb upgrade command. Maybe this was the issue? Maybe, I switched to the previous version branch, but installed the latest version of files with the "upgrade" command? — I don't really know. But when I switched to 1.16.x and upgraded the forum again, then it worked as expected. The forum now works fine.