What happens when I tell ATC to "mark the tape"?
These days, with digital recording, it's likely that nothing special happens.

"Mark the tape" is a holdover from when ATC recordings were actually on tapes - you could ask a controller to "mark" a tape to hold it and prevent it being recycled for another recording while your request for a copy of the tape is being processed. (Anecdotally I've been told some facilities had tape rotations as short as 15 days, and a FOIA request letter could easily take longer than that to make it to the facility).

Today much (probably most) of what is said on the frequency is recorded digitally, direct to hard drives, and kept around "essentially forever" as one of our local controllers put it when this came up at a recent event.  Basically the virtual "tapes" stick around for as long as the facility's recording machine has disk space to store them, and since audio files aren't that big facilities can store recordings a lot longer than they did in the days of tape.



All that's really needed to pull a record to review it with a controller/supervisor is the date, time, facility, and frequency (and your aircraft callsign would probably be helpful too) - e.g. "April 4, 2014 - JFK Local Control for Runway 04R (119.1) - United 123 Heavy".  

"Mark the tape and give me a phone number" survives as non-standard phraseology that lets the controller know that you will likely have a bone to pick with someone when you get on the ground and gives them the opportunity to tell their supervisor that an annoyed pilot will be calling the facility in the near future (and hopefully give the supervisor time to listen to the tape).
The useful part is really getting the facility phone number so you know who to call when you land.