I am using some lookup functions in multiple cells of a spreadsheet.
Depending on the search a cell may actually have a null value. When a
null value is returned the cell is populated with a '0'. My question
is how do I make it so a null value is simply an empty cell?

One way is to use
=IF(your formula = 0,"",yourformula)

Maybe I'm missing something, but isn't it possible that zero is a valid
result of "formula" that needs to be treated differently from a null value?
For example, one might be computing an AVERAGE() over a range of these
results.

I'd suggest
=IF(your_formula="","",your_formula)
instead.