These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Fridge Brilliance: Why is it Tony changes every time he goes into the mirror? As noted below, the real reason is because Heath Ledger died prior to completing the movie, but there's an in-story justification as well: Tony's a liar and a con artist, so his appearance keeps changing to match the situation. Foreshadowed by the drunk in the beginning.
Fridge Horror: Valentina has sex with Tony in the Imaginarium — which is "facilitated" by Dr. Parnassus. He doesn't seem to always know what happens inside the Imaginarium, but it's still squicky. Basically, she has sex inside her father's head.
"Funny Aneurysm" Moment: Tony is introduced hanging from a bridge; unfunny now that Heath Ledger is really dead. Just hearing this line from Tony (although he's Depp in this scene) seals the deal:
"Nothing is permanent, not even death."
A bit more subtle, but still there; the song "We Are The Children of the World" which is sung at Tony's fundraiser near the end is a satirical slash at the song "We Are The World", which was co-written and performed by music/Michael Jackson. This didn't come off as much in good taste given the pop star's death prior to the film's release. The Reveal doesn't help, since Jackson's charity work may have covered monstrous activities too.
Also, given the amount of grief the tabloids gave Heath Ledger, before and after his death, this was a humdinger:
"Don't believe everything you read. Especially The Mirror."
Harsher in Hindsight: The first time we see Heath, he's pretending to be dead. While he never actually dies on screen his character does.
Say whatever you like about Tony Shepherd but the man certainly knows how to swing a crowd!
Mary Sue: Arguably, Tony, but deconstructed once his back-story appears.
Misaimed Fandom: The song "Children of the World" was Gilliam's stab and parody at Michael Jackson's song "We Are the World", but eventually won an award to Gilliam's chagrin.
Moral Event Horizon: Throughout the film, Tony comes across as likeable, despite hints at a dark and shady past that likely involved defrauding a charity. After it is revealed that Tony's charity harvested the organs of Third World children and sold them off to rich Western buyers, his facade cracks and shows just how nasty he really is.
So OK It's Average: A rare positive example - several critics praised Gilliam for being able to salvage a good, if not great, movie despite Ledger's death, a situation where most directors would have given up... which in itself makes the movie a Crowning Moment Of Awesome for Gilliam.
Squick: Despite sixteen being the age of consent in the UK, some viewers might be a tad uncomfortable with the much older Tony wooing and eventually sleeping with a teenage girl. True, Lily Cole was 21 at the time of filming, but the in-context cradle robbery is still intact.