Deader Than Disco: Poor Buttons has gotten this nowadays. Back in his short-lived popularity, his animation projects were wildly popular and there was an explosion of fanart, memes, and shipping with Sweetie Belle. But after Hasbro's legal entanglements with creator Jan Animations, the enthusiasm quickly sputtered out after it was clear the creative force behind the project wouldn't be making any more animations. He still has his fans, but he's now descended back to being one of the many supporting background ponies.
Also seems to go into a very vocal Hate Dom on Derpibooru, due to the trope above, resulting with..debates.
The scene where Button plays Pong has had it's rounds of image macros.
A much known crueler meme is Button suffering a lot to the point of universal Chew Toy due to the cease and desist by Hasbro, to the point that is the joke. Some people really don't think it's that funny, especially with Jan having gotten the finger by Hasbro.
Pet the Dog: After the cease and desist drama dies down, Hasbro does this. The choice of video is strangely appropriate in a way.
Ship Tease: Brainfreeze notwithstanding, Button's outing with Sweetie Belle looks very similar to a date.
It was referred to as a date on the Ask The Crusaders tumblr, which is made by the same guy that made the video. It was the person asking that called it that, but Sweetie Belle didn't object. Also, it has been clear on there for quite some time that Sweetie Belle has a crush on him.
This also leads into another project from Jan Animations, "Don't Mine At Night" (A song/video parody of Katy Perry's "Last Friday Night", where the two are playing Minecraft together (as a play on Sweetie Belle's voice actress Claire Corlett's own appreciation of the game).
Tear Jerker: Meta example: Despite Jan's optimism towards Hasbro and him making negotiations, Hasbro still declined the decision to allow Jan to use Button Mash, resulting with the series being cancelled one episode in. Despite this, Jan thanked everyone for their support, and Button still has a strong fanbase, despite some due to the Deader Than Disco status.
The Woobie: Very, very little goes right for poor Button Mash, though it's played for comedic effect rather than for drama.