I think there is much evidence to show that God does weep with His people. First, Jesus is God and we know that Jesus wept. Also Jesus said, 

John 5:19

Jesus gave them this answer: “I tell you the truth, the Son can do nothing by himself; he can do only what he sees his Father doing, because whatever the Father does the Son also does.

So in other words when Jesus cries it reflects what is going on in the Father’s heart as well. And we know that the Spirit, "intercedes for us through wordless groans" (Romans 8:26). The Holy Spirit is intercedes, prays for us in inexpressible ways. Wow!

God loves us deeply and is moved to tears when we hurt. He is truly Emmanuel—God with us.