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.)
The universe is currently believed to be flat, and therefore open. The data from WMAP hows the universe to be flat to within 0.5%. I think strictly speaking "open" means $\Omega < 1$ rather than $\Omega \le 1$ but the flat universe certainly isn't closed.
If dark energy behaves like a cosmological constant then I don't think it's possible to observe acceleration in a closed universe obeying the FLRW metric. However little is know about dark energy and it has been suggested that it may vary with time. Andrei Linde has suggested that the acceleration may reverse in time causing a Big Crunch (sorry the link is a bit vague but I couldn't find anything definitive about Linde's suggestion).
Later: courtesy of Google see http://arxiv.org/abs/astro-ph/0301087 for an article by Kallosh and Linde about the Big Crunch.
Later still: it looks as if no-one else is going to answer, so I'll add a note. Assuming Luboš is right (and he knows vastly more than me!) you can get acceleration in a closed Friedmann universe, so the fact we observer acceleration doesn't necessarily show the universe is open. However, as mentioned in my first paragraph the WMAP data shows the universe is flat so we don't need to observer acceleration to know it isn't closed.
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.