If Shiftry uses "Feint Attack" against a Pokémon that used "Smokescreen" the previous turn, would it have to flip for Smokescreen or not?
Yes it would have to flip. The effect of Smokescreen lies on the Shiftry, and Feint Attack only ignores effects on the DEFENDING Pokémon
Source: PUI Rules Team (2004-01-08)

Relevant cards

Note: All applicable cards may not be listed below.
BW2_65
HGSS3_25
PL3_1
EX2_22