It seems like others have already given the definition, but I do want to introduce you to the wonderful thing that is Wiktionary.

https://en.wiktionary.org/wiki/haul_ass

It's part of the same project as Wikipedia, and it has a very comprehensive compendium of definitions for slang terms.

It means 'to be putting lots of effort into moving fast'. Often with a connotation of doing so to an irresponsible extreme.

It means to move with all possible haste.

It’s an urgent “hurry” maybe similar to “careening”

To me it means to move quickly. Often it means to move quickly under strain, but not necessarily.

(Moving slowly under strain is not hauling ass.)

The best example I can think of would be the insane running animations of Peppino from Pizza Tower:

And this scene from Rat Race (2001) is maybe less helpful but still relevant: https://youtu.be/PzjLImF8Zbc?t=1m5s

To hurry or leave.

But my mental picture is a donkey in a trailer...

In my mind, hauling ass means to go as fast as possible, generally dangerously so. There usually is a solid reason for it, like escaping something, making it somewhere on time when you're already late. I'd say it's under strain, but the strain isn't that you're carrying a load, it's that you're pushing your vehicle or yourself to its limits, or perhaps beyond them.

Some additional nuance that I didn’t see yet: there’s a specific meaning to the command or the future tense of hauling ass, as that usage tends to imply the need for haste, specifically to hurry, e.g., “we gotta haul ass”. However, in the past or subjunctive tense, it tends to mean great speed without necessarily implying anxiety or haste - the speed could be entirely innate and casual - e.g., “they were really hauling ass” or “this thing can really haul ass”.

It means all of the above, but the typical usage is to go really fast.

I have used that saying to mean time to leave.

Funniest thing I ever heard my father say when I was a kid was when we were at a stop light waiting for an overweight man to finish crossing the road: "If that guy ever wanted to haul ass he'd have to make two trips."

Obviously not something I would perpetuate or use as a considerate grown up, but provided maybe to lend some American context around the phrase.

Go fast or hurry up.

Going fast.

I thought it was similar to the phrase "quick, fast, and in a hurry" or "I need this done by yesterday"

Like it's a call to action to save your ass

You should also try to understand how this instance works including the rules.

It means 'to hurry.'

I feel like in addition to what was said, something or someone which is "hauling ass" seems to tend to be on the larger side categorically? A person who carries a lot of momentum when running (not just cuz they're fat but could also be with a big pack on) or a larger or louder vehicle. It feels like it comes with an implied "caution!" And not necessarily just because they're not being cautious as others have said.

It's more like "escape" or "leave suddenly".