Need two cells, for example B1 and B2. In B1 enter the number you want to exclude. In B2...
Formula: copy to clipboard 
=SUMIFS(A1:A10,">0",A1:A10,"<>"&B1)


PS: Earlier I stated, "If you want to pad the result for slightly early checkin or slightly late checkout the 12 in my formula represent hour of the day..." To be more precise, it represent hours until next day for checkout time, and hours from previous day for check in time.