It's not settled law, but IMO probably not, I don't believe it's ever been tested in court.
Windows comes with a click-wrap contract that you agree to when you install it, that restricts certain things you could usually do. So it's a bit of a bad example... but I'll use it anyways.
What does (generally, not legal advice, not a lawyer) need a license is making a copy of the software, i.e. installing windows onto your computer actually makes a copy, or making a copy of the installer for your friend - makes a copy. Just booting up the machine that already has windows installed on, that's (probably) not copyright infringement even if you don't have a valid license. It might be breach of contract depending on what contracts you have agreed to though! Physically giving your friend the computer with windows already installed on it, is similarly not copyright infringement. Cloning the hard-drive so you now have two hard-drives with windows installed, that's copyright infringement.
I don't think it quite works that way: the license is really just an agreement between the user and the rights holder that they will not sue for infringement. So use without license really means "you may expect to be sued". The outcome of that suit would determine whether or not you "need to license to use code". I'm guessing there have been many cases where the copyright owners for some code sued someone for using it and won.
Using software is not one of the enumerated rights reserved to a copyright owner. Copying and distributing it are. So it's unclear whether a copyright owner has standing to sue you for using their software without a license. In the US there is conflicting caselaw, hinging on the legal status of the temporary copy made in the computer's RAM when a program is launched.
For distributing it, making copies of it, and violating contacts, definitely. For using a program already on a computer with no click through contract that you agree to to run it (or other contractual obligation), I'm not aware of any cases.
That they can sue is a non statement, you can sue for anything, but I don't believe it is obviously (or even likely) copyright infringement...
To be clear, it would be ChessBase that would be sued, not individual users. And ChessBase would be the one allegedly in violation of "distributing it, making copies of it, and violating contracts". Since their license to use Stockfish seems to be revoked, it seems like it would fall under copyright infringement at that point (although up to the courts to decide, of course).
Even if you could execute the code you somehow obtained, but for which you've somehow lost the license, for ChessBase it becomes worthless, as they can no longer distribute - whether sell or give away - anything building up on it.
They do also have a cloud offering - you pay them to evaluate a chess position on their computer - so I could see the license to use the code being important, though it's separate to their flagship offering of shrinkwrap software.
Like, if I ran Windows on a server and Microsoft revoked my license. Surely that's tested in a court at some point?