Mr Socks, from Commander Kitty, is a Ferret whose native language sounds like either "Dook Dook Dook" or "HRBLRUHBLRUBH" (or some variation of those letters). It's unknown if all inhabitants of Ferret speak like this, or if Mr Socks is unique in this way, though Ace at least seems to understand him.
In Kukuburi, the big white creature in this comic speaks in electronics symbols, but it doesn't seem to hinder him.
In The Order of the Stick, one of the principal characters is traumatized and finds herself unable to speak intelligibly for 150 strips. It's represented in the text by writing her dialogue in a Cryptogram, with a different translation key for every episode. All of her lines still make sense if you can read them, leading to a rather long-running Bilingual Bonus.
The Pyros in Cuanta Vida are muffled by their masks, and it's very hard even for their teammates to actually understand what they're saying.
All major characters in Listening to 11.975 MHz are this. It isn't because they all speak foreign languages (or mathematical equations), it's because translating them gets you gibberish. The secondary characters are not much better.
Parodied in The Adventures of Dr. McNinja — Judy, the gorilla receptionist, can speak very fluently in sign language, which is subtitled for the readers but which the doctor doesn't understand a word of. She also speaks Gorilla, but it just sounds like grunts and stuff, even though it's a full-fledged language. A flashback story show she used to have a sign language translator that spoke out her signing in English, but it worked so horribly she obviously decided that even being the Unintelligible was better.
Everyone and everything in Grumble is written in an unreadable language cipher. Even the sound effects.
In Something*Positive, one of Rory's friends, Sherri, initially spoke in letters too tiny to read. Starting in her second arc, however, they're small but intelligible. As of 2016 (and following a bit of a character redesign for the gang because "apparently kids grow"), she just talks normally.
Nebula: Ceres' thought bubbles are jumbled gibberish; it's impossible for anyone, the other characters or the readers, to understand what they're saying. It's heavily implied that this (among other things) is because of Black Hole's influence.
In El Goonish Shive, the Writer's Block communicates almost entirely in weird noises. It did say one intelligible word once, but that word was censored.
Writer's Block: Mee moo f@&$.
Dregs: The nudist understands English, but cannot speak it. Instead she communicates almost exclusively in variations on the sounds "heh" and "bup".