What do we mean when we say it today?
In our world today, this phrase means to show people your real feelings. Someone who wears his heart on his sleeve is open and honest; the complete opposite of the cunning, devious hater, Iago.
There's never any hint of deceitfulness when this phrase is used either. Basically, we don't use Iago as a guide when saying this phrase. When we say this today, chances are, the person is actually honest and transparent—maybe even a bit of an oversharer.