The Black Sea is a brackish body of water, that used to be separate from the Mediterranean. Further back, it was likely part of the Mediterranean before the last ice age. How long would it, or for a body of water similar to it, take to turn from brackish/saltwater to freshwater?
It would overflow...
As of right now, the Black sea is overflowing with water, leading it to push more water out to the Mediterranean than it receives. This was not necessarily the case during the last Ice Age, when it was a lake. But precipitation patterns have changed since then, so the Black Sea could not be a lake any more.
The Bosporus has an unusual two way flow regime, where the upper layers of the water flow south, towards the Mediterranean, while the lower layers flow north into the Black Sea. According to Gregg and Ozsoy, 2002, the northward flow is roughly 11,000 m$^3$/2, while southward flow is 16,000 m$^3$/s. This means net flow in the Bosporus is a little less than the Ohio river in the US, or about twice the flow of the Rhine or Nile rivers. That is a not-insignificant amount of water. If you tried to seal up the Bosporus, the Black Sea would simply fill up and overflow elsewhere.
...into a Mega Lake...
On the other hand, suppose that you are Prometheus and you completely replace the Bosporus with a ~250 meter ridge of hills connecting mountainous Turkey and Bulgaria. With no southern exit, the Black Sea would overflow in the lowlands to the North and East. But nearby, there is a big patch of land that is below the level of the Black Sea.
The most likely path for overflow would be for the Black Sea to flood up the Don, across the Manych Depression, and into the Caspian Basin. In this case, the mega-lake would be expanded into particularly dry areas. The Caspian Sea does not overflow, because the eastern half of it is in desert, and it evaporates as much water as is poured in. At some point the Black Sea would become stable with inflowing water counterbalanced by evaporation.
...but it will still be salty.
The Caspian Sea is salty. Most endoheric seas, those that don't empty into the ocean, are salty. Without water leaving them, any dissolved salts taken into the lake will be concentrated there as evaporation removes the water the salts were formerly dissolved in. Taken to the extreme, with a lake isolated for a long time and a desert to evaporate the water quickly, you get the Great Salt Lake or Dead Sea, which are both saltier than the ocean.
So the real answer to your question, is that if the black sea was disconnected from the Mediterranean, it would never become freshwater, it would only become saltier.
A model for this situation is nearly provided by the Baltic Sea. An alternative model is provided by the Dead Sea.
Numerous rivers, accounting for most of the rain catchment areas of Eastern and Northern Europe, flow into the Baltic, while the only outflow is via the Skagerrak between Denmark and Sweden. Evaporation also occurs, but rather less than would occur further south, in warmer climates.
The flow through the Skagerrak, though it's rather wider than the Bosporus, is almost entirely unidirectional, which is highly unusual for a sea connected to the world's oceans. About once per decade, a combination of spring tides with a storm surge causes a brief reversal of this flow, hence from the North Sea into the Baltic.
As a result, the Baltic Sea's salinity is roughly 0.5-0.8% (actually about twice that in the depths), compared to 3.5% for ocean water in general and 3.8% for the Mediterranean. Baltic Sea water is thus sufficiently fresh that you could drink it without dehydrating yourself.
Conversely, the Dead Sea lies significantly below sea level and has no outflow - except for evaporation. It is not directly connected to the world ocean system. Evaporated water leaves its salt content behind; this is the principle behind the distillation method of water purification. As a result, the Dead Sea has a salinity almost ten times that of ordinary seawater.
Therefore, you can see that if the Black Sea were dammed so completely that it had no outflow at all, it would become more salty, not less - but if it were only dammed sufficiently to ensure it always flowed outward rather than inward, it might grow less salty. However, it is unlikely ever to reach zero salinity even in that case.
The answers above don't consider the possibility where the Black Sea is disconnected in the sense that it receives no water from the ocean, while still having outbound rivers flowing into it. In this case, the volume of incoming water flow I would be split between outgoing water flow O and evaporation E:
I = O + E
Now suppose the salinity of the incoming water is x and the salinity of the Black Sea itself is y. In the state of equilibrium, Black Sea would receive as much salt as it would lose. Evaporation doesn't carry any salt away, so
I*x = O*y y = (I/O)*x
So, for example, if half of the incoming water evaporates, the Black Sea would be twice as salty as river water in the long run.
The answer is that this would never happen.
If anything, water from the Black Sea would evaporate, thus concentrating the remaining water and make it more salty. By contrast, salt has no way to leave: it might be deposited by receding water but would redissolve if the water level rose again.
Of course the Black Sea is fed by several freshwater rivers, which might indeed decrease the overall salt concentration, but that would never lead to the whole lake becoming freshwater: the original salt content would still be there. As mentioned in another answer, the only way of diluting that salt content indefinitely would lead the Black Sea to overflow, thus reconnecting with an ocean.