Having another non-Tor browser open will deanonymize you if it is correlated with the information the other browsers/programs provided. This means that you should avoid visiting the same websites, using the same accounts, etc.
Assume that you have two accounts on a website, a public account which you access with a regular connection and a private account which is only accessed through Tor. Let's say your public account has your identifiable information while your private account does not.
The person who owns the website and other people can see how your public account and private account are always connecting at the same time, but never interact with each other. Thus, it wouldn't be hard to correlate the two accounts and so you will be exposed while using Tor.
There are other suspected ways where your anonymity could be compromised such as a website measuring your normal internet traffic and estimating your tor traffic speed (with the hops) and getting the estimated speed right if you visit the website, or a the supposed idea of cookie extraction. However, timing attacks are the strongest reason to avoid using another non-Tor browser at the same time.