Is the prayer of the Imam and those behind him complete if the Imam forgot to recite Al-Fatiha in the first Rak'ah and no one corrected him, and does the Imam bear the burden of that sin and the sin of those praying behind him?

Reciting Al-Fatiha is a pillar (Rukn) for the Imam and for the one praying alone. If someone omits it out of forgetfulness and a long time passes, he must repeat the prayer. Similarly, the congregants must repeat their prayer if they did not prompt the Imam or leave him after he omitted Al-Fatiha.