I've never tried Framing the Frame, but I've heard from several people that it pretty much has to be stealthed. Is that true? If so, are there any other jobs that are difficult or impossible to complete once the cops have been alerted?
You don't need to stealth any mission, but there are two missions that are not worth doing any other way:
Framing Frame is not worth it without stealth. Day 1 takes much longer without stealth and is a lot harder that way, too. On day 3, you get no gold without full stealth (no alarms, even escaping!). Without the gold, the money you get is not worth the time put into the mission.
Big Oil is notorious for being almost impossible if the intel from day 1 is destroyed and alarm goes of on day 2. On the second day, it is important that the scientist does not have time to destroy the computers in the lab. With an alarm, the scientist is likely to destroy the displays.
As of the 29th of May, there is now the Shadow Raid mission that must be completed through stealth only*. The objective of this mission is to navigate a warehouse and steal loot (drugs, money, artifacts, etc.) and return them to your van, all without being detected.
***** Once you trigger an alarm on this mission, a 1 minute countdown timer starts before the mission ends. During this period, it is possible to fill your van with the required amount of loot and escape to successfully complete the mission, however it will not count as completed through stealth.
You don't need stealth to complete any mission. Overkill decided to leave it open to players to chose if they want to stealth or not. That being said, there are missions that are extremely hard without stealth. If you don't stealth days 1 and 3 of Framing Frame, prepare yourself for a damn hard fight.