1. **They Break Immersion**
Several people will think this is stupid since "a MOBA has no and needs no immersion". However, League has really managed to convey the lore and the identity of each champion, even to the players who don't actually sit and read all the stories or background lore. Τhe gameplay has always managed to feel connected with the story, and the things that ruin that sensation are minimized. When I play a champion, their lines and animation/FX really make me feel what they're all about. Mastery emotes also disrupt this a bit, but they're so neutral that you don't think about it twice. Same with event emotes- those even enhance the immersion sometimes (_VS Event_ emotes, for example). However, being able to spam a ton of "fun" caricatures, possibly unrelated to your champion, will really drown out the subtler "in-character" interactions. Right now, the gameplay actually draws attention to the lore with intelligent, lore-rich lines, animations, interactions etc.
But who's gonna listen or care about what the enemy Urgot has to say to our Teemo when they can just spam funny emotes at each other in-game?
2. **They Make Jokes and Taunts Obsolete**
In continuation of the above, right now if I want to smack talk my opponent(s) without using the chat, the best way to do it is to use a Taunt, Joke, Laugh or Dance. All of them are carefully animated and feature multiple champion interactions that are actually voice-acted.
That's a big part of what makes champions feel dynamic and alive, and a big factor of what makes new champions and skins feel worth their cost (IP/RP).
Emotes are so much more easily digestible, however, that literally nobody will have the patience to make their Camille say or hear her hilariously smug banter ("My boy, nobody cares about your silly little country") when they can just spam a Jinx face sticking her tongue out.
3. **They _Will_ Be Tilting**
Anyone who has played games that limit all communication down to a few emotes (Hearthstone, Vainglory, Clash Royale etc) knows that even the most neutral and rudimentary of expressions ("Thanks", "Sorry", "GG", "Angry") can very easily turn into infuriating, passive-aggressive psychological jabs.
The innocent "Jinx sticking her tongue out" emote for dodging a hook can easily start to feel very abusive when the enemy Lee Sin uses it _every single one_ of the nine times he ganked your lane and killed you, even if it's not actually spamming or anything like that.
Riot has officially stated that things like "EZ" at the end of the game are considered abusive. Do you think seeing an infuriatingly "fun" crying Amumu face every time a fed enemy destroys you is somehow not?
**Taunts are NOT actually abusive because it's one champion taunting another. It's just part of the story. Emotes, however, are taunts aimed _directly_ at the players by design.**
1. **Sometimes They Can Enhance The Experience**
On the /dev post for the Emotes, it was written that "Emblem" emotes were considered and actually rejected because they don't convey much. And yet, some of the best emotes so far have been exactly that. Team Ice/Team Fire, Chaos VS Order, the silly yet beloved URF emote. All of those make the player feel part of something, they're fun to use but they're not abrasive enough to be abusive, because they're not personal.
When I got killed by a Dawnbringer Riven as Nightbringer Yasuo during the event, and got the blue emote flashed in my face, it felt like Order was gaining ground against Chaos, not like _xXxRiv3nOTPxXx_ was saying I was trash. I actually think that LESS expressive emotes will be a much wiser choice here. They feel more immersive, less inflammatory and they still feel good and fun to use (when GP deletes your champion with a barrel crit, even a generic "Bilgewater" emote can really get the point across :P).
2. **They Make People Type Less**
This is pretty self explanatory, but usually it's easier to use Ctrl+5 than to type "lol no0b" to your lane opponent. Even the generic Mastery emote has actually substituted a lot of unnecessary typing. It can be used for banter or intimidation, to show off after a good play, to lightly taunt an enemy that made a mistake or admit that you messed up with the hilarious posthumous mastery flexing. :P
To conclude, personally I don't think that Emotes are terrible idea, however their implementation needs VERY careful handling in order to make them an actually fun and welcome addition to the game, instead of ruining immersion, lots of animation/voice acting work and giving players an easy way to behave badly.