This is the complicated formula Ali M refers to. It's an array formula entered with ctrl-shift-enter:
=IF(COUNTIF(A2:F2,"<>0")=0,0,SUM(A2:INDEX(A2:F2,SMALL(IF(A2:F2<>0,COLUMN(A2:F2),""),MIN(3,COUNTIF(A2:F2,"<>0"))))))
Note that it works if there are less than three non-zero values.