How to use regex replace in Postgres function?

Instead of fixing clause after the fact, you could avoid the problem by using concat_ws (concatenate with separator):

clause = concat_ws(' and ', "name='john'", "age='24'", "location ='New York'") 

will make clause equal to

"name='john' and age='24' and location ='New York'"

This can be even simpler. Use right() with a negative offset.

Truncates the first n characters and you don't need to specify the length of the string. Faster, simpler.

Double quotes (") are for identifiers in Postgres (and standard SQL) and incorrect in your example. Enclose string literals in single quotes (') and escape single quotes within - or use dollar quoting:
Insert varchar with single quotes in postgresql

Since this is a plpgsql assignment, use the proper assignment operator :=. The SQL assignment operator = is tolerated, too, but can lead to ambiguity in corner cases.

Finally, you can assign a variable in plpgsql at declaration time. Assignments in plpgsql are still cheap but more expensive than in other programming languages.

DECLARE
   clause text := right($$and name='john' and age='24' ... $$, -5)

All that said, it seems like you are trying to work with dynamic SQL and starting off on the wrong foot here. If those values can change, rather supply them as values with the USING clause of EXECUTE and be wary of SQL injection. Read some of the related questions and answers on the matter:
http://stackoverflow.com/search?q=[plpgsql]+[dynamic-sql]+EXECUTE+USING

concat_ws sounds like the best option, but as a general solution for things like this (or any sort of list with a delimiter) you can use logic like (pseudocode):

delim = '';
while (more appendages)
    clause = delim + nextAppendage;
    delim = ' AND ';

If you want to do it with regular expression try this:

result = regexp_replace(clause, '^and ', '')