In my testing, 0 only occurs if 0 is the result of the formula - Subtracting TODAY from TODAY.
So, to EXCLUDE a 0 result change your formula from >= to > ....this will solve PART of the problem.
array_constrain(MIN(FILTER(O2:X2; O2:X2 > 0));1;1)
If all the results were 0 or UNDER (a negative number) the output result of the formula is #NA.
Therefore, to handle the #NA simply wrap your formula in IFNA() like so...
=IFNA( TODAY() + ARRAY_CONSTRAIN(MIN(FILTER(O2:X2, O2:X2 > 0)),1,1) )
However, the MIN function returns a single value which is what you are attempting to do with ARRAY_CONSTRAIN(). So you can remove it.
=IFNA( TODAY() + MIN( FILTER(O2:X2, O2:X2 > 0) ) )
That should do it nicely.
If it doesn't work let me know but I tested it fine.
Cheers,
Keith