I'm presuming these mixer taps are not 1/4 turn ceramic disc types.
Sounds as though the washer has stuck itself down onto the seating. Depends on the make of the tap, but the washer is probably mounted on a small brass plunger that is pressed down onto the seating when you turn off the tap. Turn on the tap and the washer plunger relies on water pressure to lift it off the seating. The other possibility is that the washer plunger actually screws up into the shank of the tap handle, and this has somehow or other come adrift.
I'd try turning off the HW supply - there should be either a service valve close to the tap, or a shut-off somewhere close to the boiler - and taking the top off the tap. Check that the washer plunger is free to move. It would also be worth making sure you have a replacement washer before you start!
HTH.