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Some ideas:
 
"The Conversation" and "Blow Out"-- obvious ones.
"Testament"-- the mid-80s nuclear holocaust movie, the doomed husband's
only appearance is on an answering machine.
"Dressed to Kill" (and probably other DePalmas) is big on tapping. For example,
"Bonfire of the Vanities"...
If you'll include video recording as well, there's Egoyan's "Speaking Parts"
and of course "sex lies and videotape"...
A different sort of "recording technology" in Wenders' "Until the End of the
World" (dreams)...
A dead character leaves a message on video for the hero in Tarkovsky's
"Solaris"...
A recording of the initial scene is replayed several times through "Blade
Runner"...
An answering machine is central to a painful sequence in "Short Cuts"...
 
Hope this is helpful!
Andy