The exponential map is not open on the von Neumann algebra of all bounded linear operators on an infinite dimensional separable complex Hilbert space. A 1987 article by Conway and Morrel shows that the spectrum of an element of the interior of the image of the exponential map does not separate 0 from infinity. On the other hand, a bilateral shift U has spectrum equal to the unit circle centered at the origin, and every Borel logarithm on the circle applied to the unitary operator $U$ yields a preimage point for U under the exponential map. Hence U is in the image but not in the interior of the image.
I learned about Conway and Morrel's article from this answer by David Speyer.
