I found this post after experiencing exactly the same thing. I experienced it at other times over the past couple of years but my patience was greater then and the problem seemed transient.
Being less tolerant now, I have carried out advice elsewhere in these forums and disabled half the addons, the problem continued so I re-enabled and disabled the other half and so on, with each problematic half until finally down to two add-ons; Dugi and ElvUI. Disabling either appears to stop the ‘bug’, so something is going awry in their interaction with each-other and the WoW Client.
Obviously, the fault could be in ElvUI, but everything seems fine in its interactions with the other 30+ add-ons I use (which I suppose means nothing) but… (a) ElvUI has no setting or code that I know of for moving or equipping gear, (b) this was the only post of its kind found by Google and (c) we gotta start somewhere. 
I have looted many more things, keeping them in my bags to see if it did it with them (it did not), and moving the ‘bad’ gear to other slots and bags. The affected gear is as follows:- (Tankadin, Lvl 20+ when spotted, 25 when investigations began, and 28 on making this report.)
1) Sparkmetal Bracers (iLvl 30) gets replaced by Lightsoul Vambraces (iLvl 25)
2) Dread Pirate Ring (iLvl 32+) gets replaced by Ring of the Fool (iLvl 22)
I have watched the bags while fighting and, obviously, it cannot and does not happen during the fight but seconds after, presumably when interested add-ons scan the bags for new gear to act upon. I witnessed the exact moment of change once at about 1.5 seconds after the loot dialog disappeared.
There may or may not be arguments for the equipping of either or both but I need to know that I’m gaining max XP when levelling and that my decisions regarding the non-Heirloom slots aren’t being compromised, which is why – when this started – I turned off the Gear Advisor altogether. I normally value and welcome it’s appearance on screen. To have gear auto-equip (without even the on-screen comparison to alert me) when it is turned off is very frustrating. 
I hope you can find and fix this, as I love your add-on. (While you’re at it though; one minor request, can you get the Gear Advisor to completely ignore potential Heirloom replacements unless the character has outlevelled it?)
Many thanks for your many hours of hard work and polish.