Here's how I got online play working:
TL;DR:
- Install the game from Steam. (Origin or DVD might work, but I can't say for sure.)
- Run it.
- Cancel the login window.
- Enter Paradise City and watch the super long intro movie.
- Go to the menu, find Online Options and go to Account Management.
- Log in. Important: DO NOT select the option to remember your account!
- Have fun online!
- The next time you play, repeat steps 2 through 7 (sans unskippable intro, which is thankfully a one-time-only thing).
Full story:
Burnout Paradise Network (I think that's what it's called) doesn't work, at least on my machine (MacBook Air late 2013, 11", OS X 10.10, Crossover 15). If it pops up, the screen goes black, and you have to use taskmgr (from Run command) to quit Burnout. The Network pops up if you log in when you start the game, so don't; instead, go through the in-game menu when you're past the account loading and all that. You will be able to play online, and the Network is not going to pop up if you log in through here. Now, if you let the game remember your password, the next time you start it, Burnout will auto-login you, trying to load the Network and crashing again. If you've done this, go to the savegame directory and delete the auto login file. Then do it as explained above.
Hope this helps somebody! 😊