Does logging a SLOW query impact postgres performance?
How is the production system reporting the query times now?  If you are already getting that data, it is questionable that getting it a separate way will be helpful.

But to answer your question, there are two kinds of overhead you will get.  

Setting log_min_duration_statement to any non-negative value will cause the system clock to be queried more often.  This should be almost negligible on modern common hardware and kernels, but can be quite slow on older things.

The other overhead is actually writing the log entries.  If the server is very busy, this is probably going to be quite noticeable with log_min_duration_statement=0.  You can probably come up with some threshold greater than 0 will that still log the slow statements, but without logging every single statement.