Yes and no-- yes, a normal Barbarian would, but no, yours won't.
Barbarians are naturally only proficient up to Medium Armor. So under normal circumstances, if you hadn't given the Barbarian a third feat in Armor, they would take a DEX penalty to AC (and possibly other penalties), based on the fact that they don't know how to wear the armor, move in the armor, etc. Basically, they're clumsy with it, however 3rd Edition defines "clumsiness".
However, a Fighter does have the Heavy Armor feat already, thus is trained in using it, so even if s/he duals to Barbarian, s/he can wear heavy armor without penalty.
Afaik, that's the only possible penalty you could encounter in that situation, unlike if you were an arcane spellcaster (the heavier the armor, the higher the chance of spell failure), monk (wouldn't get Wisdom bonus to AC), or scout-type character (wouldn't hide well, or for very long).
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