Scans showed a fracture in the foot which Mullen injured during Sunday's goalless derby with Hibernian.
The 19-year-old, who made his debut in the Scottish Communities League Cup semi-final over Inverness, had been in line to play in Sunday's Hampden clash against St Mirren.
Mullen told Hearts TV: "Gutted is an understatement. The injury couldn't have come at a worse time, but I'll have an operation this week hopefully and then it will be all about getting back fitter and stronger for next season."
The teenager joins Danny Grainger, Marius Zaliukas, Callum Paterson and Scott Robinson in suffering a premature end to his season.