You want to use SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2,
criteria2], …)
Also you forgot to use Indirect when referencing your sum range in the begining of your formula.
Use this formula below, if cell B1 contains the sheet name and cell A5 contains the Date to compare.
=SUMIFS( INDIRECT( B$1&"!$D$6:$D$60" ) , INDIRECT( B$1&"!$E$6:$E$60" ) ,"<="&$A5 , INDIRECT( B$1&"!$F$6:$F$60" ) , ">="&$A5 )
I hope this helps.