dampen vibration
increased structural integrity
help absorb light impacts so frames dent or ding less
dampen vibration - foam would do that. So will cabon fiber or titaium with no added weight.
increase structural integrity - Actually, a hollow tube is much more resistant to bending than a solid tube. Bikes are not bending in half so adding material is not needed.
help absorb light impacts so frames dent or ding less - if you hit it hard enough to dent the metal. Lightweight foam is not going to help. It would have to be very dense, ie heavy, foam to provide any protection.
It could also trap moisture in any voids.