I started getting much better results when I started putting the LUT last or at least after one color corrector. Imagine you used EI 1000 (or just "expose to the right") to have cleaner shadows... If you apply a generic LUT first, you'll blow out your image's highlights immediately. No amount of color correctors after the LUT will ever bring that information back. Putting one color corrector first, however, lets you darken your shot and have greater control over your highlights, saturation, etc. before the LUT is applied.In theory, it should be LUT first.
The way I see it, the LUT is sort of amplifying everything from that flat image. It boosts saturation, deepens the blacks, boosts the highlights, etc. So if you happen to have a rich red in your shot (maybe a light, a jacket, etc), there is a possibility that the LUT will actually cause that color to clip. Putting a color corrector Pre-LUT would let you dial back the color before all of that amplification happens.
For a while I was applying the LUT first and it just gave me headaches because my footage wasn't shot absolutely perfectly. Figuring out the "CC first, LUT second" approach totally changed that.
I should also mention that programs like DaVinci Resolve even support this usage built into the system with Input LUTs and Output LUTs. They even point out that an Input LUT can destroy information that can't be recovered in later nodes.
