How about a few examples?
Sometimes, you can eliminate failure points by cutting away material so that load is diverted elsewhere in the structure. So technically, that's correct. But in general, and specifically with barrel flutes, removing material
1. weakens the barrel (both in bending and in hoop stress).
2. makes the barrel less stiff (it will bend more when you push on the end.
3. makes the barrel vibrate at a slower rate.
4. makes the barrel vibrate *differently*. Flutes can introduce a bias to certain directions (this can be good or bad and is not explored much for some reason).
1 is not a big deal unless you go crazy with flute depth, at which point it becomes a very big deal.
2 is also not a big deal, except that it is the cause of 3, which is a small deal because it changes the way the rifle moves when fired. I will not go as far as the above video that J_530 posted that says stiffer is always better, but generally, that would appear to be the case. Sometimes you want the right stiffness, not just more stiffness. 4 could be a big deal at the hyper-accuracy end of things. I could see asymmetric barrels helping with long range benchrest, for example. But you'd have to work out what you were trying to accomplish and how.
The video is also a little obtuse about the heat dissipation. While it's clear the author understands what's going on, he gets a little lost in detail. Surface area DOES speed up heat removal from the barrel. But not that much. Flutes are effectively very poorly designed cooling fins. If you wanted cooling fins, you'd make them annular, not length-wise, and you'd want lots of thin ones of a pretty significant height. It would be very George Jetson (and awfully impractical). The bigger effect of fluting losing mass, which causes the barrel to heat up more with each shot, but even that is pretty small.
All in all, flutes are a strange obsession in the shooting world. There are other factors with the same or greater effects that are hardly ever talked about. Like barrel length - nobody ever worries about the decreased stiffness (and increased heat dissipation) due to a longer barrel, but for flutes, it's all of a sudden a big conversation.