Does the current acceleration of universe imply that our universe is open? If the universe is closed, from the Friedmann's equation, the acceleration of universe wouldn't be possible, would it be? (Of course, except for the very earlier time of inflation era.)
No. Universe that is accelerating can be open, closed, or flat in principle. The latter attributes are determined by how much matter/energy is in the universe relative to the critical value. If the energy is greater than critical then the universe is closed, if it's less than critical the universe is open, and energy equal to critical the universe is flat.
On the other hand, the condition for the universe to accelerate is for it to have a component with negative enough pressure (in technical terms, this is entirely obtained from the second Friedmann equation that gives the acceleration of the characteristic scale of the universe, which is not required when talking about flat/open/closed).
[That said, it happens that current cosmological data strongly favor a universe that is flat. But this is mostly independent of the statement that the universe is accelerating. ]
Without dark energy open/flat (closed) universe implies one that is expanding forever (ends in a Big Crunch). But with dark energy that drives the acceleration of the universe, this "geometry is destiny" link is broken. In particular, the future of the expansion can be arbitrary, as it depends on the future behavior of dark energy. However, if the dark energy continues to accelerate the universe as it is doing now, then the universe will expand - and also accelerate - forever.
If "open" means nonpositive spatial curvature (which is the usual meaning), then no. The experimentally measured spatial curvature is consistent with zero, but the error margin includes values on both sides of zero. There's no conflict in the Friedmann equations between accelerating expansion and positive spatial curvature.
If "open" means "never recollapses", then the answer is still no. There are models consistent with the Friedmann equations and all experimental data in which the universe eventually recollapses (see http://arxiv.org/abs/astro-ph/0409264). In these models the dark energy is not the cosmological constant, but a scalar field similar to the inflaton. (The problem of explaining the end of inflation is similar to the problem of constructing a model in which the future exponential expansion ends.)
There are other ways in which the universe might be closed in either sense. If future data constrains the spatial curvature to a negative value, it's still possible that we're in a region of unusually low density in a universe that is overall spatially closed (although this is no different from saying that the actual value of the curvature might be outside the error bars). It's also possible that FLRW cosmology is wrong (you assumed it is correct in the body of your question, but not in the title).
I hope @ron-maimon will explain why he said the answer is yes.