Is there anyway I can set up a rule to not allow overset text in a text frame? I want to print without getting the "Overset text" alert for an specific text frame.
Object → Text Frame Options → Auto-Size (tab) → Auto-Sizing = Height Only (or both)
This solves your issue.
If you want hide a part of the text, just put it into a frame (Paste Into). This frame became a mask for a text box.
You can setup such behavior for any new text box as default. This can be done from 'Objects Styles' panel.