RSS reader for iPhone with text-to-speech support?
For RSS, I use Reeder 3, but you can probably use your favourite app, because…

iOS has built-in text-to-speech support system-wide for selectable or unselectable text.

Method 1: Select text and speak it

Open Settings → General → Accessibility → Speech and enable Speak Selection.


  A Speak button will appear when you select text.


In an app like Reeder, hold down on the text, make a selection or choose Select All, then tap Speak.




Method 2: Speak all text

Open Settings → General → Accessibility → Speech and enable Speak Screen.





  Swipe down with two fingers form the top of the screen to hear the content of the screen.


A HUD appears for controlling speech, which hide automatically or with the left arrow and unhidden with the right arrow.