Does the Tanya say a tzaddik Gamur is infallible?
Do other groups claim their leader is a tzaddik gamur?
Do Lubavitchers maintain that every Lubavitcher Rebbe was a tzaddik gamur?
@Freddie is a Tanya expert, not me. Maybe he can chime in.
I don't think the word infallible is used, but it's clearly the intention that for a T"G, sinning would be impossible.
It's clear that it's talking about a once in a generation tzadik, which is what all chasidim beleive their Rebbe represents, the leader of their generation.
So yes, Lubavitcher chasidim always felt their Rebbe was the leader/T"G/Nosi/Mashiach of that generation.