From my limited experience, soft tires feel more bouncy than hard ones, with same pressure. That is, if you lift and drop front side of unloaded bike, with soft rubber on front wheel it will jump like a bouncy ball (kind of). But with hard rubber it'll jump more like badly inflated football/basketball (at low pressure) or like a sack of potatoes (at high pressure).
Other things being equal (size, tread, pressure), do you think soft rubber tire will provide better suspension than hard rubber?
EDIT: made a test with Schwalbe CX Comp (soft) vs some Vee Rubber harder compound tire (same width) - dropped front side of unloaded bike from a small height. CX Comp definitely feels more bouncy and jumps more silently. But actually maybe they bounce with same rebound, since it's hard to measure precisely.