Why do Catholics leave Jesus crucified on the cross in their churches when Jesus is risen?
Jesus is not "left on the cross" but rather his saving love is displayed. The central message of a crucifix is that Jesus died for our sins out of love. The centrality of this to the mission of Jesus on earth is mentioned several times in the New Testament:
For Jews demand signs and Greeks look for wisdom, but we proclaim Christ crucified, a stumbling block to Jews and foolishness to Gentiles (1 Cor. 1:23).
“I am troubled now. Yet what should I say? ‘Father, save me from this hour’? But it was for this purpose that I came to this hour" (John 12:27).
And just as Moses lifted up the serpent in the desert, so must the Son of Man be lifted up, so that everyone who believes in him may have eternal life.” For God so loved the world that he gave his only Son, so that everyone who believes in him might not perish but might have eternal life (John 3:14-16).
Then he took the bread, said the blessing, broke it, and gave it to them, saying, “This is my body, which will be given for you; do this in memory of me.” And likewise the cup after they had eaten, saying, “This cup is the new covenant in my blood, which will be shed for you" (Luke 22:19-20).
God could have chosen any method for the redemption of humanity. In his desire that humanity know the depravity of sin and the depths of God's love, he chose to die on the cross. Displaying a crucifix does not in any way deny the resurrection of Jesus. After all, do faith communities that display only a cross want to downplay the sufferings of Jesus? Of course not, it is just a different preference of emphasis. In the Catholic Church, the crucifix has been the traditional sign and statement of our faith in God's love and mercy. It reminds us that Jesus never promised his followers an easy life (Matt. 16:24) but that if we join our sufferings to his, we too will ultimately triumph as he did.