No. That is the point of a portable application. It does not use the hard drive/file system of the host operating system.
Some of your data may, however, remain in the RAM/memory of the host system until it is overwritten, or the system rebooted. A capable hacker could possibly extract them, if he knew what to look for in all the gigabytes of garbage.
I should say you are pretty safe with your portable browser on a USB stick in all normal cases and environments, but if the owner of a host system is a) your enemy or b) a very capable hacker or both, and c) you have some really classified top-secret stuff to protect, then don't use that host system for maximum protection.
