The answer is yes they do.

On a flight back in 2010(/), I was on the flight deck of an A340 coming into land and we were maybe 200 feet off the runway when the pilots hit the go-around button. The reason was that a mobile phone (which turned out to belong to a member of the crew) caused interference on the VHF audio circuits meaning that ATC instructions were missed. On a B747 passengers are immediately above the avionics bay and their phones could cause a similar problem. I have also seen unmanaged WiFi cause interference on aircraft.

Also, a mobile phone at 10,000 feet can see many Base Stations, several of which will be using the same frequencies and whilst logged on to one base station the phone - which will have ramped up to maximum power will cause interference on several other base stations - and that causes teh mobile operators problems.

In reply to a post by MHC:

OFF means OFF, completely off, not standby. On a PC, hibernate is fine as that is off with the last state stored, whereas standby turns off the screen and drives, reduces teh CPU speed but keeps the memory active.

Any electronic device has the potential to disrupt aircraft systems, be it an ebook, PC, phone, portable radio ... There are RF emissions from every devices - normally quite low but still have the potential and if a component has failed the device may still operate but cause even greater problems.

I hope you turn your mobile off every time you visit a petrol station

"What about my digital watch? It's an electronic device and doesn't have an off button."

Non-Bond digital watches don't have several watts worth of RF transmitter, so it's almost infinitely improbable that there'll be any trouble from an ordinary digital watch.

Also, modern digital watches intentionally have very low power electronics in them so that the batteries can be small and low powered and still last a bit longer than they did in the days of (eg) Mr Sinclair's Black Watch.

Lots of modern consumer electronics does have a watt or twos worth of RF transmitter, powered by quite chunky batteries The flight crew are not experts in what does and what does not have an inbuilt transmitter, so to make life simple the request (rule?) is "switch it off" at times when there is least wiggle room (takeoff and landing).

The chances of a problem occurring as a result of something being accidentally or deliberately left on are small but not zero. The more such devices that are on, the greater the risk of Bad Things happening.

Please, if you wish to play Russian Roulette, do it when and where it affects your life only. Otherwise, switch it off. Thank you.