If I'm running the Tor browser and Google Chrome at the same time will my Chrome browser affect the anonymity of my Tor browser?
Chrome browser uses an own binary, safes its configuration and other files in a special location. So all things that Chrome does is divided from other software in your system especially from the Tor Browser Bundle. So Chrome does not manipulate a Tor Browser Bundle and the TBB does not manipulate Chrome.
However if you're using Chrome and the Tor Browser Bundle for the same sites and you show the same browsing behaviour the site might be able to connect your sessions. But this is not an issue of Chrome or other browsers, but more about your surfing habits.
TBB uses its own configuration. (Firefox settings are independent from the system.) Chrome uses the default system settings. Those two won't go into each other's way, or even know their counterpart exists.