You can only make a conversion on the same webserver, the only thing that can be different are databases, as long as you can connect to both of them.
The converter only pulls data from the forum and imports them into a phpBB3 forum. It doesn't change anything on the old forum.
In your case I would do this: First make a test conversion. Set up a clean install of phpBB3 on your new webserver and transfer a mirror copy of the old site to it too. Then run the converter and lead it to the mirror test copy. See how the transfer goes and fix any errors. If everything goes well, close your live forum and make the conversion again, so you have consistent data. Then continue with your new phpBB3 forum.