Six games against the Boston Bruins in 2011/12, zero wins for Toronto. The Leafs held a lead for a total of 21 minutes over the six games, or a mere 5.8% of playing time. It's difficult to get a lead while scoring 10 goals and surrendering 36 over six contests.
Somewhat surprisingly, this seasons' series with Boston was only the second time in franchise history where Toronto was swept outright in a series of at least six games. For all the terrible teams that the Leafs iced during the 1980's, they always managed at least a tie in their season matchups. Way back in 1925/26, while still called the St.Patrick's, Toronto lost all six games to the Montreal Maroons and in the process were outscored by a collective 25-13.
I checked through the hockey-reference database and found the following season series of at least six games in which Toronto didn't manage even one win; (W-L-T, GF-GA)
- 1985/86 vs. Minnesota 0-7-1 31-47
- 1981/82 vs. Minnesota 0-5-3 19-33
- 1976/77 vs. Buffalo 0-5-1 13-28
- 1972/73 vs. Montreal 0-5-1 12-25
- 1926/27 vs. Ottawa 0-5-1 4-17