Homestar Runner. This is lampshaded in the eponymous flashback in the Strong Bad Email flashback; it features crudely-animated art based on that in The Homestar Runner Enters the Strongest Man in the World Contest, a children's book made by the series' creator before their website was established. In the Email lady...ing, Strong Bad presents a clip parodying the crude animation style and personalities of the very, very early Homestar Runner short Marshmallow's Last Stand.
Fairy Foxes has gone under Art Evolution throughout its 6 episodes. The foxes started out as cute little chibis. However, their designs changed to an anthro look and stayed that way since episode 3.
And since episode 7 came out, StickFreeks (the creator of Fairy Foxes) has switched from MS Paint to Adobe Photoshop Elements 7.0.
Much like the Fairy Foxes example above, Wolf Dragons also started out different than it is today. Here are examples of the pilot episode: Before and after. Additionally, voices didn't appear in the first version until around episode 3, while in the better version, there were voices from the start.
Red vs. Blue had a unique form of evolution: through the first few seasons, the voice acting and video quality improved somewhat, from the low-quality early episodes to much better sound and video by Season 4 or 5. In addition to progress with equipment, though, the series gained significant upgrades with the release of each new Halo game, and by Revelation, also features extensive CGI thanks to Monty Oum joining the crew. It comes full circle with the release of remakes of the first 5 seasons, re-filmed in HD.
It was stated at one panel that as their sound equipment gradually became better, fans started to complain that the voices sounded "too good" and the team decided to work the audio a bit to maintain the low quality feel.
Somewhat justified in that the characters are supposed to be speaking either through communicators inside their helmets or through their helmets, so sounding crystal clear is rather odd.
Nameless progressed from relatively crappy quality to quality rivaling that of some professionally-made cartoons in a few episodes.
The BIONICLE Flash episodes (which consisted of cutscenes from the Mata Nui On-Line Game and a short series of non-game animations) went through some degree of evolution:
At first, the animations, as well as the game, blended 3D graphics with standard, 2D Flash animation, and as such, some of the movements had many individually rendered frames showing off bodyparts/machinery from just slightly different angles. The characters also had a lot of shading, and the backgrounds were immensely detailed. As the game progressed, the visuals changed to standard 2D imagery, with the animation consisting of the same couple of images being repositioned to simulate movement. The characters only had a few basic angles to their parts, and barely any shading that matched their environment, which also became less detailed. So when you went back to the first areas of the game, characters suddenly tended to look out of place. Thus, this can be seen as an example of the art de-evolving through the course of a series/game.
The Kanohi Pakari, Great Mask of Strength (the type of mask worn by Onua Mata). It went through several redesigns, even though the actual LEGO piece itself was not all that complicated. When Onua first appeared, his mask looked really close to the real piece, with some minor differences. By the end of the game, he and every other Pakari-wearer were outfitted with a completely Off Model-looking mask, that only faintly resembled the original. When the web episodes took over from the game, the animators thankfully settled on a design that later became standard, and looked exactly like the Pakari.
An Akatsukis Life. From its somewhat crudely animated and muffled-sounding beginnings, it eventually evolved into this. The latest episodes contain a handful high frame-rate shots that (aside from the obvious pixels) wouldn't look too out of place in an actual cartoon show.
One could say the art style for Neurotically Yours went through this, starting from very rough design and animation and it now a lot smoother.
The art in Happy Tree Friends became less stiff and more "cute" over time, mainly by making them overall a little rounder and cartoonish. The overall animation quality also improved. See this illustration◊.
Ashley in The Crazy Kids of Grade 5 has went through a character redesign. Compare this ◊ to this ◊.
Cow of the Wild has gone through this over three years. In the first episodes, the feet were large in comparison to the rest of the body, several wolves had a lot of facial hair (i.e. Alari's bangs), eyes were one color with white highlights, and cat characters were frequently mistaken for wolves, or vice versa. In the more recent episodes, the feet have gotten a bit smaller, the eyes are multicolored, and there's a distinct difference in the basic character design of different species. The lineart and shading have also become much more refined.
While it isn't a leap by the standards of other examples due to the simplistic style, there is a noticeable jump in quality from episode 1 of DSBT InsaniT to episode 2 onward.
The music videos for Savlonic, a fictional band created by "Weebl" Picking, have undergone this—the third and fourth videos, created by another animator, are far more advanced graphically and no longer depict the characters in borderline Super-Deformed style.
Turnabout Storm's art and animation quality saw a few improvements after the fist two episodes for different reasons:
Ace Attorney characters, who use their in-game sprites, went from the pixel-ly sprites of the Nintendo DS releases to the much smoother artwork of the iPad's HD re-release whenever they are compatible. Besides that, the dialogue box's font was changed to the same one used in Ace Attorney.
While the Ducktales characters still look as crappy as they did in the first Ducktalez, Vegeta has seen an incredible improvement between Episodes 3 (fairly well animated for a Flash cartoon in that age) and Episode 7 (professional-quality CG animation).
GEOWeasel goes from very Flash-influenced animation with same-width pencil outlines to a very messy but more varied style in episode 9 that eventually evens out through the rest of the series.
Many Garry's Mod users tend to do this as time goes by and they learn the craft more. Notable examples include RubberFruit, Dr. Face, and Eltorro Rus 64, all of whom started off with standard sketchy animation and eventually worked their way up to something you'd expect from a big budget stop-motion film.