Ideally, you shouldn't have to "memorize", and playing the changes to jazz standards should come just as naturally as playing the changes to Happy Birthday.
This kind of skill is achieved by playing songs by ear, finding the chords by ear - or "by heart". If you can play I, IV, V type songs by just sensing and feeling how the harmony swings to either side of the tonic (I like to think that V is on the left side, IV is on the right side and I is in the middle), you extend that capability by feeling the other direction as well - swinging to the "dark side", the minor side. I has vi as its corresponding dark version, IV has ii and V has iii ... and then there's III which takes you very strongly to the dark side. You learn to feel these swings by finding chords to songs by ear, and practicing to feel difference between things. This is done by playing more and more difficult songs by ear.
The difficulty hierarchy of distinguishing harmony swings (changes) could something like this, in order of increasing difficulty: (YMMV and feel free to have a different opinion)
- left/center/right (V - I - IV)
- major/minor side, i.e. whether it's a IV or ii ("dark side")
- seventh chords - is it maj7 or dominant seventh
- diminished and augmented chords
- common special tricks like minor iv instead of IV (I think there's a name for this)
- common modal alterations like Dorian and Mixolydian mode
- inversions - the basic swing is e.g. left/right, but there's some additional twist mixed in. For example C/G - G7 - C is a common pattern. Can you identify that C/G when you hear it, and reproduce it? How about first inversion, C/E?
- "mixed" chords like F/G (like G11) and G/F (or G7/F)
- common modulations like half-step up, or to parallel minor/major
- other jazz tricks like tritone substitutions
- special uncommon modal feelings and chords like the Steely Dan chord, e.g. G/C, highest note B
- playing completely outside, layering an "alternative harmony progression" on top of other chords
These are learned one by one, and as you learn to identify and reproduce each new difficulty level, it becomes a part of your own vocabulary. After that, playing a song becomes more like telling a story or a joke - you remember the big picture and the point of the joke, and if needed, you can improvise the details if needed, it doesn't have to be "memorized".
(Btw, if someone has lists of songs ordered by difficulty along steps like those listed above, each song introducing only one new harmony trick, please post a link.)
However, even after you've learned all this, there's still one added challenge, namely actually remembering the tune. ;) Some songs are just so entirely forgettable, un-melodic, or weird that you have to write them down. Song structures are often like that too - is this the second or third chorus now? Were we supposed to do one or two rounds of the bridge here? But if you can hear and reproduce harmony, it helps tremendously, because even if you don't really remember the song, you can listen to what others do and float over it. ;)