Facebook has not responded to requests for comment about the picture, shared to celebrate Instagram reaching its 500 million monthly user milestone.
FBI director James Comey has previously said he also covers his laptop’s webcam to prevent hackers spying on him.
And digital rights group the Electronic Frontier Foundation (EFF) said it regularly sold its webcam “stickers”.
Documents leaked by former National Security Agency contractor Edward Snowden allege US and UK spy agencies intercepted webcam images from millions of Yahoo users around the world between 2008 and 2010.
And, in 2013, a BBC Radio 5 live investigation found sites where hackers exchanged pictures and videos of people captured without their knowledge.
Hackers attack webcams via attachments containing malware.
If these are opened by a user, a hacker can gain remote control of the functions on their computer, including the webcam.
Security company Symantec warned users not to keep computers with webcams in their bedrooms and “not to do anything in front of one that they wouldn’t want the world to see”.
Prof Alan Woodward, a computer security expert from the University of Surrey, said: “There is so much malware that can access these physical parts of a computer and observe and listen to users.
“It makes absolute sense to put tape on the webcam.
“In fact, I don’t know why manufacturers don’t sell laptops with an in-built slider to cover it.”
But taping a microphone made less sense, he said.
“Sounds can travel through tape – it is just another membrane.
“So it begs the question whether we need a physical switch to disconnect the microphone.”
Recent research suggests hackers are now exploiting the accelerometers in phones to pick up sound.
