None too horrible come to mind, although I wasn't entirely happy with aspects of the Monkey Island 2 HD re-release. (I know the "HD remaster" label also covers gameplay remastering, but I'll focus on visuals and style here.)
Although MI2 was originally rendered in glorious 320x200 VGA resolution, the visuals were mostly "serious" and pseudo-realistic, which provided a nice contrast to the zany, uh, everything else in the game. The intro demonstrates this nicely, starting out quite ridiculous, before segueing into what would've been a very serious Ye Olde Pirate Game title sequence, if not for the occasional interruption of Guybrush chasing dancing monkeys across the screen. All of it scored with somber and dramatic state-of-the-art MIDI music, of course. (The aspect ratio is off in the video, as is often the case with captures of 320x200 VGA, but it's close enough.)
The game keeps shifting tone between silly and serious all the way to the very ending, which honestly would be pure nighmare fuel if not for the constant string of zany antics.
Now, the HD remake includes a completely new intro sequence, which keeps the monkey chase (in original VGA quality, too) but drops the serious parts, even hamming things up a bit with silly voice over guy reading the title card "Deep in the Caribbean, Scabb Island". You can hear the actor wiggle his fingers dramatically... and it's just too much, like they're afraid we might already have forgotten that this is supposed to be funny, and need to remind us.
Of course, MI2 is notorious for its darker tone, a direction which was promptly reversed in MI3... and now, to some extend, also MI2 HD.
Oh, also, the HD release removes Khris Brown's cameo. How dare they.
Now, for an example of a perfect HD re-release – outside gaming – look no further than Blade Runner: The Final Cut, as compared to the previous Director's Cut: Generally improved imaging, color balancing, a few special effect shots cleaned up (but not dramatically changed), small but annoying continuity errors removed. The most noticable difference is a few scenes of gore, which had previously been toned down, are toned up again, but nothing that changes the overall tone or style of the film. And Deckard still shoots first.