No, why should it?
Any program you install can contact the Internet, if you don't like the Safari web browser (and many don't) then your certainly free to install a alternative.
Just don't install a web browser from AppStore, because Apple has a 4-8 week delay while they review all apps there, leaving a window of opportunity for your machine to be hacked.
Apple is notoriously slow in issuing security updates for OS X, Safari, iTunes etc, doing so many months (or in one case 3 years) after being notified of the issue.
To install Firefox you'll likely have to head to Apple Menu > System Preferences > Security > General and where it says Downloads select "Anywhere" (you can always turn it back later)
Then visit this page, check the status of your plugins and there also is a download for Firefox on the same page.
It will be in your Downloads folder and you open that, double click the installer or package to open it to show the program and drag it to your Applications folder, then from there drag a icon (alias) to the Dock so you can access it quickly.
Firefox has tons of Add-ons, more than anyone else.
Plus tons of persona's that dress up the browser
It's a customizable browser, there are already many themes others made
It also auto updates by itself and the add-ons, security issues are fixed very rapidly and can come out within hours of a exploit being known.
So it gets updated very fast, unlike Safari which is glacial.
Oh, there is a tiny chance if you have a lot of add-ons with many tabs opened and closed, that memory management is a bit hairy with Firefox and you might not be able to access the menu items at the top.
It's gotten better in more recent versions than in the past, but if it occurs, simply close all open browser windows and that will allow access to the menu and you can create a new browser window.
Other than that slight issue, Firefox is absolutely the best browser on any platform.
Wonderful thing about Firefox, it runs/updates in the permissions level of the user account using it, even in Standard user.
This makes it a lot more secure in a sense as it doesn't have access outside the Standard user account, like in Admin or root level.
Chrome is auto-updated via root and this has caused issues. Root should be offlimits to nearly all third party software for security purposes.