Try =IF(AND(D5<37256,D5>37226),D5,0)
(or =(IF(AND(D5<37257,D5>37225),D5,0) if you mean to include the start and end of the month.)
D5>01/01/2005 means D5>1 divided by 1 divided by 2005, which it isn't normally if it contains a date.
Excel stores dates as integer numbers, starting from 0 (1Jan1900). Understanding that makes working with Excel dates a lot easier. (Times are the decimal portion, so a date of 0.5 is 12 midday on 1Jan1900.)