The TTS isn't great here, I agree that it does sound like the R just disappears, but that is not what should happen. I couldn't find tallrikarna on Forvo, but you can hear the same sound in pojkarna as said by a native speaker here: http://sv.forvo.com/search-sv/pojkarna/
The R melts together with the N to form a sound that is often called retroflex. The same or similar goes for T, D, L and S.
We don't have a really good post about this yet as far as I know, but there's some information here: https://www.duolingo.com/comment/6235538
Is there a rule for the plural of tallrik? I can't seem to find one in the tips & notes for this section. It only mentions that en-words with the stress on the last syllable usually take "er". Is it usually "ar" for those which are not covered by the rules? Or do I have to memorize the plurals for all the words which are not covered by the rules in the tips & notes?