In addition to @frank-van-puffelen's answer, I do not believe that Firebase RTDB should itself cause your game to lock up until a network connection occurs. If your game is playable on first launch without a network connect (ie: your logic itself doesn't require some initial state from the network), you may want to check the following issues:
Make sure you can handle null. If your game logic is in a Coroutine, Unity may decide to silently stop it rather than fully failing out.
If you're interacting with the database via Transactions, generally assume that it will run twice (once against your local cache then again when the cache is synced with the server if the value is different). This means that the first time you perform a change via a transaction, you'll likely have a null previous state.
If you can, prefer to listen to ValueChanged over GetValueAsync. You'll always get this callback on your main Unity thread, you'll always get the callback once on registration with the data in your local cache, and the data will be periodically updated as the server updates. Further, if you see @frank-van-puffelen answer elsewhere, if you're using GetValueAsync you may not get the data you expect (including a null if the user is offline). If your game is frozen because it's waiting on a ContinueWithOnMainThread (always prefer this to ContinueWith in Unity unless you have a reason not to) or an await statement, this could ValueChanged may work around this as well (I don't think this should be the case).
Double check your object lifetimes. There are a ton of reasons that an application may freeze, but when dealing with asynchronous logic definitely make sure you're aware of the differences between Unity's GameObject lifecycle and C#'s typical object lifecycle (see this post and my own on interacting with asynchronous logic with Unity and Firebase). If an objects OnDestroy is invoked before await, ContinueWith[OnMainThread], or ValueChanged is invoked, you're in danger of running into null references in your own code. This can happen if a scene changes, the frame after Destroy is called, or immediately following a DestroyImmediate.
Finally, many Firebase functions have an Async and synchronous variant (ex: CheckDependencies and CheckDependenciesAsync). I don't think there are any to call out for Realtime Database proper, but if you use the non async variant of a function (or if you spinlock on the task completing, including forgetting to yield in a coroutine), the game will definitely freeze for a bit. Remember that any cloud product is i/o bound by nature, and will typically run slower than your game's update loop (although Firebase does its best to be as fast as possible).
I hope this helps!
--Patrick
