I have an Xbox One and an Xbox 360 with the same Xbox Live profile on both consoles. If I were to sign in on the Xbox One and then sign in on the Xbox 360, would the profile on the Xbox One automatically log out or am I be able to sign in with no trouble?
As a matter of fact, yes. You are able to be logged into one Xbox One and one Xbox 360 at the same time.
You can be signed in to one Xbox One console and one Xbox 360 console at the same time. There are some apps that you can't use on different consoles at the same time. For example, Groove Music can only be used on one console at a time.
However, if obtain an Xbox One in the future, you are not able to log into your profile on multiple Xbox Ones.
You can only sign in to one Xbox One or one Xbox 360 console at a time. If you try to sign in on a second console, you'll be asked whether you want to sign out from the first console. If you respond yes, your account will be immediately signed out of the first Xbox One console.
Source: Xbox Support - Concurrent SignIn