The primary risk with explosives is losing a yellow timed-drop reward by getting the killing blow on an enemy with one - similar to losing weapon fragments in the first game. (Not all enemies will drop rewards at all, and killing one which would have triggers a 'loot destroyed' popup on the lower left, so if you don't mind save-scumming you can entirely avoid this and still blow up ADVENT left right and center.)
There are also supply raid missions which give an explicit objective to 'not destroy critical supplies'. Central calls out specific cover items the first time you encounter them in one of these missions - similar to the 'please don't break that' messages for Power Cores/Flight Computers in EU/EW - which look like some kind of glowing crate in a metal frame. I presume that they function similarly to any other cover object in terms of being destroyed (i.e. both explosives and stray weapons-fire can potentially break them), and further presume that any which are destroyed will decrease your end-of-mission reward. However, I've never actually blown one up to test that.
Otherwise, there appears to be no downside to reckless explosive use. XCOM has actually grown more hard-hearted about such things over the past two decades: these days, the maps haven't had all the civilians conveniently evacuated beforehand and collateral damage is still not a concern...
(Incidentally, I believe it was merely a rumour that explosives reduced UFO rewards in EU/EW (other than the actual object-on-the-map items like Flight Computers, Power Sources, etc.). I can't find a solid source for that, but personal experience and the wiki both indicate that they didn't reduce alloy/elerium rewards in the Long War... and I find it hard to believe that vanilla was more hard-core than the mod.)