I've got a file said to contain information I was looking for. Unfortunately it is an executable instead of DOC (as it was meant to be) and the site I've download it from looks suspicious for me. If I was not using Linux, I'd run it on a VM or a separate PC. But running Linux, do I need to worry, or can I just run it with Wine? Can Wine system be infected?
The news is only slightly better than it is on Windows. If you run the executable, it can do anything an executable can, up to and including getting whatever data is in your home folder.
The good news is that most Windows viruses aren't (yet) written in a way to work well on Wine. In some cases you can delete the wine folder and be otherwise unaffected.
The bad news is that a wine executable is an executable in the full Linux sense -- there's nothing stopping it from doing anything a malicious shell script might, including escaping the .wine folder.
Wine has a wiki page on securing Wine here: http://wiki.winehq.org/SecuringWine -- partial measures you can take include things like scanning a file with ClamAV before running it.
There was an ancient article about this 5 years ago. Short version: Almost all Windows viruses aren't written to run well on Wine. Maybe Wine has gotten better, but it's not the sort of compatibility that Wine is looking to offer.
There are a couple viruses that infect both Windows & Linux but they're very very rare and didn't spread all that well.