vlookup displays "0" if reference cell is blank?

=IF(OR($C$4="",$C$4="No Match"),"",VLOOKUP($C$4,Comments!$A$2:$E$295,5,0))

First off let me point out that the above is really no different to your earlier formula just shortened slightly and lookup range reduced to A:E given you're returning value from E you need not reference anything beyond that.

To answer your question... if it's purely presentational (as it would seem) you're best bet would be to use a Custom Format on the cell as opposed to double evaluating the entire VLOOKUP, eg:

[=0]"";General

So 0 will persist in the cell but the cell will appear as blank... this is preferable to doing the VLOOKUP twice over.