It is noteworthy that Ramadan is a blessed month and an opportunity for every Muslim to get closer to Allah, increase his or her faith, and gain abundant rewards and forgiveness. A Muslim is recommended to do as many good deeds as possible in Ramadan,  because all good deeds in this blessed month merit greater rewards.
In this context, Prophet Muhammad (peace and blessings be upon him) is reported to have addressed his Companions on the last day of Sha`ban saying,

O people! A great month has come; a blessed month in which is a night better than a thousand months; a month in which Allah has made it compulsory upon you [Muslims] to fast by day, and voluntary to pray by night. Whoever draws nearer (to Allah) by performing any of the (optional) good deeds in (this month) will receive the same reward as performing an obligatory deed at any other time; and whoever performs an obligatory deed in (this month) will receive the reward of performing 70 obligations at any other time. It is the month of patience, and the reward of patience is Paradise. It is the month of charity, and a month in which a believer’s sustenance is increased. Whoever gives food to a fasting person to break his fast will have his or her sins forgiven, and he or she will be saved from the Hellfire, and he or she shall have the same reward as the fasting person, without his reward being diminished at all. (Reported by Ibn Khuzaymah)

In his response to this issue, Dr. Monzer Kahf, a prominent economist and counselor, states,

It is not obligatory to pay zakah during Ramadan. It is, however, better to benefit from Allah’s reward bonuses that are given in Ramadan for good deeds. Allah (Glorified and Exalted be He) has special times and places for extra reward bonuses. Good days for making good deeds include Ramadan, especially the last 10 days of that month, and the first 10 days of Dhul-Hijjah up to the first day of `Eid Al-Adha. This is why many people prefer to adjust their zakah time to coincide with the last 10 days of Ramadan.

Sheikh Ahmad Kutty, an Islamic scholar,  also states,

It is not obligatory to do so; however, it is recommended if one does, for all good deeds in Ramadan merit greater rewards as it has been stated in one of the Prophetic hadiths.