I have seen both, and now I am unsure when to use which. To the best of my knowledge "rendered mute" is roughly equivalent to "rendered speechless" and "rendered moot" to "rendered irrelevant". But I have also seen usage that contradicts this. Help?
Phrase "rendered moot," idiom:
At some point, this whole debate may be rendered moot.
(ODO, moot, adj, 2 - open 'More example sentences')
Phrase "rendered mute," literal/ metaphorical.
Some are deprived of the ability to reason and some made blind and others rendered mute.
When Jesus had cast out the demon, the mute man spoke. The crowd was amazed."
(Jesus the Messiah ..., p.182 GoogleBooks)
The use of mute in the idiomatic sense is a mistaken use of similar sounding words (that seems to be catching up of late).
Google nGram "rendered mute,rendered moot"
Oxford Dictionaries Moot Trivia:
The word 'moot' can be traced back to the Anglo-Saxon era of British history when a 'moot' was the meeting of prominent figures and nobles from the local society to discuss matters of regional importance.
Your intuition regarding the two expressions is roughly correct. There is no definition of "mute" that isn't in some way related to silence or speechlessness.
Confusion may arise from "moot", however, as the American and British usage of the adjective differs. The American "moot" indicates that something has no practical significance. This is more or less the definition you've given. The British "moot", however, indicates that something is open to discussion or debate. With that said, I'm having some trouble finding references to the phrase "rendered moot" in the British sense, so I suspect it may be an American idiom and, therefore, would indicate irrelevance.
With all that said, it's also entirely likely that the contradictions you've experienced are simply results of malapropisms - people using similar sounding words in the wrong places.