$_='|value 1|value two|"|Value 3 was "NULL"|2001/06/06|';
s/(\|.*?)(")(?!")(.*?\|)/$1$3/;
print;
[download]

there are 4 sets of parens in the regex, I'll try to break them down :
the first one looks for a pipe, then any number of characters (but not being greedy about it)
the second grabs the lone quote the third is a negative lookahead. This is where is golden nugget of regular expression goodness lies!
the negative lookahead makes sure that there's no quote following the one from the second backreference. Also, since it's a zero-width assertion, it doesn't create a backreference of its own.
Finally, the last set of parens describes "the rest of the string" up to the ending pipe-delimiter.

So,
"look for a pipe, and then any characters up to a quote, make sure it's not followed by another quote, and then the rest of the string, up to a pipe"
Now the one caveat for this re is that it will misbehave on "", but my reg-fu is not strong enough to determine the handler for that contingency.