Imajin said:
What if King Zedekiah of Judah (597-587 BC) listened to the warnings of the Prophet Jeremiah and other and did not rebell against Babylon? Let's say the Kingdom remains tributary to Babylonia until the rise of Persia. How would Persia treat Judah? Would the House of David end up subservient to yet another throne?
If the King of Judah swore loyalty to Persia, the King of Kings would probably name the King of Judah as the Satrap of the Province of Yehud. This was a common practice in the Persian Empire.
What would happen after the Persians is questionable. If Judah continued to support the Persians when Alexander arrives, Jerusalem probably suffers the fate of Tyre, the House of David is deposed, and a Greek governor appointed. If they switch sides and support Alex, possibly Alex continues the King of Judah as the local governor (as he did with many local Satraps in OTL...he tried to continue the Persian system of administration as much as possible).
They then pass to the Ptolemies and the Seleucids, who probably are going to remove the King of Judah and install a Greek governor. The Romans will certainly do so, if the Ptolemies or Seleucids do not. So by the 1st Century AD at the latest, the House of David is deposed and Judah comes under direct foreign rule.