guises: Came here to find out if anyone had had any success running it in WINE, guess I'll have to figure it out for myself.
osce: Well, this is a 6-year old thread, but just to answer your question (and hopefully helps out others with the same question), BioShock Infinite for MS Windows in GoG runs quite well in wine 10.0 without any tweaks, at least on Fedora Linux. I'm pretty sure it would work under any other Linux distributions.
I also have BioShock Infinite native Linux from Steam, and the native one is much more smooth (you wouldn't notice unless you actually compare them).
I myself finished the game and all 3 extras a while ago in native Linux.
I also finished BioShock Remastered from GoG using wine in Linux recently. I also had that in Steam, but never bothered since it was flaky to run in wine at the time. I'm pretty sure Steam would also work in wine nowadays, but at least GoG version worked fine after first install.
I'm now onto BioShock 2 Remastered in Linux under wine, and so far so good.
Since New Year I finished entire series in Wine on Arch Linux - including both original and remastered versions of 1&2. Overall they all work pretty well out of the box.
BS1 - both original and remaster worked absolutely flawless, with zero crashes during full playthroughs. The only small issue I had with original was not being able to change keybindings via the options menu because input was not correctly registered for some reason.
BS2 - in this case both original and remaster were occasionally crashing. Crashes seemed totally random although I was usually able to play for an hour or two before it happened, so it wasn't too bad. But sometimes they could crash shortly after launch as well, no rule at all. In case of the original it seemed to be getting worse in later levels.
I highly recommend saving often. Infinite - mostly flawless after some work. At first I couldn't make it run in proper full screen - my desktop's taskbar simply wouldn't hide which also caused keyboard input being not recognized in game. Enabling virtual desktop in winecfg fixed it. Then there was crackling sound out of the box. Launching the game with PULSE_LATENCY_MSEC=150 variable mostly fixed that, but it still could start crackling during longer sessions, which would get progressively worse and then only restart would help. Especially using the kinetoscope machines seemed to be a risk factor that could break sound after watching few of those. I think I had one crash near the very beginning (which could be also related to sound breaking issue) but then nothing during the entire game and both DLCs.