Don't see anything about memorizing games in books like A Guide to Chess Improvement by Dan Heisman (2010)
https://web.archive.org/web/20140708105628/http://www.chesscafe.com/text/review781.pdf
and Studying Chess Made Easy by Andrew Soltis.
https://web.archive.org/web/20140708090448/http://www.chesscafe.com/text/review750.pdf
Fragments from a 2017 discussion:
"... how many chess games can you replay (till the end) ..." - VimalKumarK
"I couldn't replay any although I could recognise quite a lot." - CM JamesColeman
"Good point. I could have replayed that one. Also Reti-Tartakower where White sacrifices his Q on d8 and then forces mate with a double check. So maybe 2. Both of those are rather short though." - CM JamesColeman
"Oh yeah, that one. Maybe 3 then ..." - CM JamesColeman
"There are many many must-know games for general ideas and themes but not sure of the practical value of having all of them memorised move by move all the way through to resignation?" - CM JamesColeman
"I don't think you have to have them memorized. Just playing them over, even fairly quickly will help put the patterns and ideas into your head." - Morphysrevenges
"Yes, agree with you entirely Morphysrevenges, but the OP was asking about memorised games that you can replay out at will. For me at least, that's not many." - CM JamesColeman
In case you are curious, here are some of the games that were mentioned in the January 7, 2017 discussion:
http://www.chessgames.com/perl/chessgame?gid=1233404
http://www.chessgames.com/perl/chessgame?gid=1250654
http://www.chessgames.com/perl/chessgame?gid=1259009
http://www.chessgames.com/perl/chessgame?gid=1224575
http://www.chessgames.com/perl/chessgame?gid=1132699
http://www.chessgames.com/perl/chessgame?gid=1132581
"... Memory is too valuable to be stocked with trifles. Of my fifty-seven years I have applied at least thirty to forgetting most of what I had learned or read, and since I succeeded in this I have acquired a certain ease and cheer which I should never again like to be without. If need be, I can increase my skill in Chess, if need be I can do that of which I have no idea at present. I have stored little in my memory, but I can apply that little, and it is of good use in many and varied emergencies. I keep it in order, but resist every attempt to increase its dead weight. ..." — Emanuel Lasker, Lasker’s Manual of Chess
Memorize Games or Positions
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I did both. For example the Morphy Opera Game, Spassky Fischer Game 6. I had about 60 on a paper list that I went through on a four a day basis.
I also memorised certain key positions as in pawn structure and pieces of the mate in four variety.
Certainly playing the Morphy game through in my mind before going to sleep at night added greatly to my four move vision when working out variations over the board.