Updates get applied immediately without staging the changes for the next restart.
The builds through the package manager can't do that.
You can have Firefox set up to check and install updates automatically and everything will be taken care of by Firefox. Everything stays in sync at all points in the update process. The Firefox builds directly from Mozilla don't have this issue because they stage updates so they only get applied when Firefox restarts.
Will it crash? Possibly! Will it access memory in an unsafe way, causing security issues? Probably! Will it be a worse experience than just telling the user to restart to get everything back in sync? For most people, probably. Anything the main process does from that point on is undefined behavior.
If Firefox gets updated while it's running and needs to spin up a new content process (or if the gpu process crashes and needs to be restarted, or an extension gets installed and needs to be run in an isolated process, etc), suddenly the running main Firefox process can end up in a situation where the files on disk telling it how to set up the new process and how it expects to communicate with other processes is different from the file running in memory. With Firefox now using multiple processes to run web content and other internal things, it's much easier for things to get terribly out of sync if the underlying files get changed out while Firefox is still running. Whenever you shut it down, the next run would pick up the update's changes and you'd use them. The running firefox process could keep doing its thing, since it was all self contained. In old times, it was okay* for your package manager to install updates while Firefox was running. The problem comes from using your package manager to do your updates. The ideal solution to your problem is the option in about:preferences#general > Firefox Updates: "Check for updates but let you choose to install them." Firefox will continue to visibly notify you when it's time to update, but it won't actually download and install anything until you tell it to. Since you're (presumably) on Linux, a good analogy is performing a large system upgrade, then continuing to use it without restarting you're going to run into some problems, be they minor things like icons appearing wrong or major hangups that make the whole system unusable.
Problem is that some of those files are in use, so Firefox can't update all of them until it closes, but since some of them have already been changed, those new files would cause problems for the older version of the browser if they were used simultaneously. There was a thread about a similar issue a while ago, but the TL DR version is that when Firefox updates, it (obviously) has to change some files.