Whenever I look back on the early episodes of Avatar: the Last Airbender, I realize that Iroh was probably acting a little ridiculous on purpose. He knows that Zuko still has fresh emotional wounds from his cruel, uncompromising father and sadistic sister, and the one source of softness and warmth in his life, his mother, is long gone. Iroh always tried to be a friend to Zuko, but now that his nephew has been scarred and banished, he tries to be goofy and funny and carefree so desperately hard because all he wants is for Zuko to smile and relax again. If making a fool out of himself is what he has to do, he’d do it a hundred times over.
He was also playing the long game. I think it is plain to see that while Iroh maintained a love for his homeland, it had betrayed him and his philosophy (his time in Bah Sing Se taught him to love other cultures and not to desire their destruction). He was in some ways, hoping to rescue his nephew from that cruel fascist ideology, which was why he shadowed the boy so protectively, but I think also, to some extent, he knew the boy’s quest to find the Avatar would put him directly in a path to become a major part of the resistance, ushered in by the discovered Avatar. He knew he was one of the strongest Fire Benders. He knew he had great strategic merit and skill. He was biding his time, SEEMING to be a silly old fool, so that the Fire Nation would allow him to wander around, seeking the Avatar, which was precisely what he wanted to do, to end the war. If Ozai had believed Iroh to still have his wits, he would have kept the old man a prisoner (as evidenced by his brief time in prison later in the series), rather than to let his greatest potential threat wander around looking for the banner of the resistance.
Iroh was a strategist. He both loved his nephew and saw the potential Zuko would have, both as a future leader and as a vehicle to get Iroh closer to the Avatar.