It is about economics. If Burger King leaves Russia, they're still leaving the restaurants, distribution networks and the employees behind. They're not going to sit idle just because the brand left, rather they're going to keep going under a new name, just like how McDonald's did.

What effect does that have on the Russian war effort? None.

But Burger King sells to Russians and repatriates some of its profits back to the US. If BK leaves Russia then Russians will spend their money on other restaurants, presumably ones which do not send profits overseas (due to the sanctions) which paradoxically means more money staying in Russia.

If the restaurants close then there will be some momentary unemployment but there's a war going on so unemployment helps smooth over manpower issues. That's also ignoring the possibility of a Russian entity stepping into BK's shoes like they did with McD's and just running the place without the trademarks.

I don't see any purpose beyond moral posturing.