Pretty often, actually, depending on where you are.
The more touristy metro stations, such as Deák Ferenc tér and Keleti pályaudvar, often feature a kontroller or two checking for tickets upon exit.
Many of the M1 stops, too, have ticket control when you leave the station. Again, you’re especially likely to be asked for your ticket in the more tourist-oriented stops. On the M1, that means all stops from Oktogon west and from Hősök tere east.
Also, you’ll have to show your ticket upon entrance to any metro station, no matter where it is. Just flash your pass to the kontroller or validate your ticket at one of the orange machines right by where the kontroller is standing.
On buses and trolleys, you’ll need to show your pass or validate your ticket to the driver in many circumstances: On the busiest downtown routes, you must enter the front door of the bus and show your pass during evening hours and on weekends. On more rural routes, you’ll have to enter through the front door all the time.
On occasion, a kontroller will get on the bus, tram or trolley at random stops and check for tickets. We’ve had it happen several times, and believe me, the fine is not worth the hassle of having to buy a pass. The fine is 8000 HUF if you pay on the spot, and 16000 if you don’t.
Just the other day, we saw a BKK employee inspecting tickets on the 4-6 tram; it was the first time in 8 months we’d seen it happen on the 4-6, but again, the fine is not worth the risk.