You can use an array* formula along the lines of:
=MIN(IF(A1:A100>0,A1:A100,40000))
assuming your dates are in A1:A100.
* As this is an array formula, then once you have typed it in (or
subsequently edit it) you must commit it with CTRL-SHIFT-ENTER instead
of the usual ENTER. If you do this correctly then Excel will wrap the
formula in curly braces { } when viewed in the formula bar - you must
not enter these yourself.
Dates are stored by Excel as the number of elapsed days since a
reference date of 1st Jan 1900, so current dates are about 39200 -
hence, the use of 40000 in the formula will treat this as a date
somewhere in the future, but you could use any number larger than
this.
Hope this helps.
Pete