From the sources that I am aware of, there is absolutely no truth to this whatsoever.

All the main sources concerning the siege state that the Turks first broke through the defences at the Milandrion part of the wall, by the Gate of San Romano. This part of the wall had been the most battered by Turkish cannon and by 29th May was held up by barrels and mud. This area of the defence was the responsibility of the defence's strongest troops, the Genoese mercenaries of Giovanni Giustiniani. When he was wounded at the height of the battle at San Romano and the Turk janissaries attacked at the same time, the defence collapsed and the defenders fled from the outer wall through the San Romano gate.

The historian Doukas mentioned that a small gate was left open in the walls, the Kerkoporta, which the defenders had used to sally out and attack the Turks. Through this gate he says that some janissaries sneaked in and could fire on the defenders from behind in this way. There may be some conspiracy that the Jews left the gate open, but from the evidence, this is completely unfounded.