When did we stop sanctifying the new moon based on testimony?
The sanctification of the new moon based on testimony is dependent on the establishment of a set Beis Din [i.e. Sanhedrin] in Eretz Yisrael. It is not dependent at all on the establishment of the Temple, and hence for many years after the destruction, the moon was still sanctified through testimony. This sanctification continued throughout the times of the Mishneh, and Talmud up until the end of the Talmudic period, during [and including] the generation of Abaya and Rava. In their generation, the Jewish settlement of Eretz Yisrael became dispersed and there did not remain a set Beis Din to sanctify the moon. From that period and on, the moon has been sanctified through the set calendar. Historically, the last Sanhedrin/Beis Din to have sanctified the new moon based on testimony was the Sanhedrin headed by Hillel Nesiah [the same Hillel mentioned above who created the calendar] in the year 359. Since that time, the new month is determined based on calculation. From the above it is understood that the original Semicha continued to exist well past the times of the Tanaaim and until at least the year 359 when Hillel sanctified all the future months. Some Rishonim learn that the Semicha was indeed nullified in the generation of Hillel. Other Rishonim learn that it was nullified a generation after Hillel.
Will we sanctify the moon based on testimony after Moshiach comes?
After Moshiach comes and the Sanhedrin is re-established, we will once again sanctify the moon based on testimony.
After Techiyas Hameisim: In the period of the resurrection, the moon will have its own light and hence the concept of sanctifying the new moon will no longer be relevant.
 Rambam Hilchos Kiddush Hachodesh 5:3; See Rosh Hashanah 21a; Sanhedrin 12a; Beitza 6a
 Machazik Bracha 527; Beis Efraim O.C. 51
 Sefer Hachinuch 4; Machatzis Hashekel 417; Derech Mitzvosecha p. 69
 Igeres Rav Haiy Gaon
 Ramban; Reah; Yesod Olam, brought in Torah Sheleima 13:6-119 [p. 97]
 Ramban in debate of with Friar Paul [Pablo Christiano] p. 87, brought in Torah Shleima ibid, that the Semicha continued for 400 years past Yoshke; see Torah Shleima there for a Likkut of other Rishonim
 Admur 473:49; Levush 421:1; Sefer Hachinuch 4 “Hillel sanctified all the months until the coming of Eliyahu”; Sefer Hasichos Mishpatim 1992; See Beitza 5b and Toras Menachem 1984 3:8251; Rambam Kiddush Hachodesh 5:1
Other sources: See Machatzis Hashekel 417 that “Hillel sanctified all new moons until the end of time”
 Toras Menachem 1984 3:1528