There are a total of twelve "collectible" cassette tapes. The game doesn't track that you collected a tape until you complete a mission, so make sure that you don't quit out mid-mission after grabbing a few tapes or else you'll have to collect them again.

Chico's Tape 5
Ground Zeroes main mission: On a dumpster behind and to the left of the stairs with the yellow railings before going down to Paz. These stairs are on the side of the Admin Building where you have access to the control box that shuts down the power. This tape only appears before you rescue either Chico or Paz.

Chico's Tape 2
Ground Zeroes main mission: In the Admin Building's watch tower, on the desk.

Metal Gear Solid V: Ground Zeroes Music Tape
Ground Zeroes main mission: In the cage behind Chico.

Chico's Tape 3
Ground Zeroes main mission: Given to you by Chico after you rescue him.

Chico's Tape 4
Ground Zeroes main mission: Extract the prisoner to be executed (Rescue achievement)

Chico's Tape 6
Eliminate the Renegade Threat Side Op: In the locked armory by the warehouses, near Palitz's walking route. Basically, head to the place where you start in the Ground Zeroes main mission, and walk straight away from the chain linked fence close to that first wooden watch tower you encountered. Make your way towards the warehouses. Look for the armory on your right, it's got a red door.

Chico's Tape 1
Complete Intel Operative Rescue Side Op.

Agent's Recording
Classified Intel Acquisition Side Op: In the Admin Building's watch tower, on the desk.

Classified Intel Data
Classified Intel Acquisition Side Op: Find the bald soldier walking around the boiler room under the Admin Building. Interrogate him and he'll drop the tape. To interrogate, sneak up behind a soldier, hold when prompted, hold to open up the dialogue options, then flick the up to deliver the "Spit it out!" line. If you have difficulty finding the bald soldier, restart the mission and make sure you head to the boiler room straight away.

Chico's Tape 7
Destroy the Anti-Air Emplacements Side Op: Rescue the prisoner where Paz is located during the Ground Zeroes mission (part of Extraction achievement).

Jamais Vu Music Tape
Jamais Vu Extra Op: Close to where you begin the mission, at the nearby helicopter extraction point on a rock. Use night vision to help see the tape and nearby mines!

Zone of the Enders: The 2nd Runner Song
Beat every mission on hard, the final rank you've earned for a mission does not matter here.

Maka has provided a video guide: