I remember learning in a shiur (sorry, can't think of the reference) that Chazal was able to essentially destroy the yetzer hora of avodah zora but with the side effect that there wouldn't be nevuah anymore. Is this true? If so, can anyone produce a reference to this in the Gemara or Midrash?
The Gemara does not mention this side-effect. It just mentions the loss of "reward" for fighting the yetzer hara of avodah zara. There may have been other effects, and it may be that this reward includes specific spiritual benefits, such as the ability to reach Nevuah. For example, I think R.A. Kaplan says that the yetzer hara for avodah zara was the desire to reach pseudo-spiritual meditative highs without the hard work necessary to reach real ruach haKodesh. When people lost this 'meditative drive' they also lost the connected 'ability' for ruach haKodesh and Nevuah.
.."And they cried with a loud voice unto the Lord their God." Now what did they say? — .. 'Woe, woe, it is that [sc. idolatry] which destroyed the Sanctuary, burnt the Temple, slew the righteous, and exiled Israel from their land; and still it sports amongst us! Hast Thou not set it before us that we might be rewarded [for withstanding its allurements]? But we desire neither temptation nor reward!'... They fasted for three days, entreating for mercy; thereafter their sentence fell from Heaven, the word emeth [truth] written upon it... The shape of a fiery lion's whelp issued from the Holy of Holies, and the Prophet said to Israel, That is the Tempter of Idolatry. Whilst they held it fast, a hair [of its body] fell out, and his roar of pain was heard for 400 parasangs. [In perplexity] they cried: 'What shall we do? Maybe Heaven will pity him !' The prophet answered: Cast him into a lead cauldron, and cover it with lead to absorb his voice, as it is written, "And he said, This is wickedness; and he cast it into the midst of the ephah: and he cast the weight of lead upon the mouth thereof"..
The best known source for this connection is the Gra's commentary on Seder Olam Rabba ch. 30: The Seder Olam refers to the cessation of prophecy at the time of Alexander the Great: הוא אלכסנדר מוקדון שמלך י"ב שנה. עד כאן היו הנביאים מתנבאים ברוה"ק מכאן ואילך הט אזנך ושמע דברי חכמים. The Gra in his commentary explains: פירוש משהרגו יצר הרע, בטלה הנבואה. This idea is often cited by R. Tzadok ha-kohen mi-Lublin in his seforim.
R. Betzalel Naor in his Lights of Prophecy, pp. 22-23, cites an earlier source in Sefer Hasidim:
When signs were being performed by the prophets of Ba'al, if the prophets of the Lord would not have produced signs, the people would have turned to idolatry. Once the inclination to idolatry was eradicated, a prophet was no longer required.