Dfyjkt

I am currently in Russia on a 90-day student visa that expires in September. I have tickets to attend the FIFA Confederations Cup 2017 final match on July 2nd. Attending this event requires registering for a FIFA FAN ID, which gives access to the stadium. According to the FIFA FAN ID FAQ:

FAN IDs are issued to all spectators of the FIFA Confederations Cup
2017 for access to a stadium. FAN ID gives the opportunity to use free
transport services on the match days in the host city the sports
events. The foreign citizens, who come to the Russian Federation as
spectators of the FIFA Confederations Cup 2017 matches, can use their
FAN IDs for multiple visa-free entry into and exit from the Russian
Federation upon presentation of valid identity documents that are
recognized as such by the Russian Federation, during the period that
starts ten days before the date of the first match and ends ten days
after the date of the last match of the FIFA Confederations Cup 2017.

Considering the above, would it be legal for me to leave and re-enter Russia for tourism purposes (e.g. visit Finland or Estonia) by using my FIFA FAN ID?

It seems to me that the intended use of this feature of the FIFA FAN IDs is for travelers to enter Russia for the purpose of watching the FIFA Cup, not for visa-holders like myself to leave for tourism purposes.

Lastly, even if this is technically legal, is it a good idea?

I'd be interested in hearing the pros and cons of such travel.

This is not a good idea. Visas and visa waivers are issues for specific purposes, and visa-free entry with the FIFA Fan ID is to allow you to watch football matches and do a little tourism, not to study. When you try to re-enter Russia using the Fan ID, the immigration officer will immediately see that you have an unexpired student visa, and will likely (and correctly) conclude that your reason for entering Russia has nothing to do with football. I would expect you to be denied entry in those circumstances, leading to a whole bunch of problems: you can't finish your course, you can't get your stuff from your apartment, getting visas in the future will be more difficult, and so on.

It looks like a valid move from you, but you probably should be prepared for some questions. After border officer scan your ID, they definitely will get all information about you, and with high probability they'll see your visa. You might be asked about it, if it still present, and do you have it with you (as if you've lost it, they need to cancel it as quick as possible). Next question will be, why aren't you using your visa. If it allows multiple re-entries, then you're fine, just losing some time at the border.

However, it's really easy for you just to use your visa, I think.

Update: as from comments, it looks like your visa is single-entry, so, it's probably not the best idea to check the system, as you could look very suspicious: FAN id is working only for 10 days after FIFA ConfCup, and your visa is till September, so, technically speaking, you're breaking the law, entering the country second time, as you clearly not going to return from Russia after FIFA Id ends.

Even if you manage it out, you still could face some troubles in future with getting other visas, as the embassy stuff will see that you've entered Russia two times with a single-entry visa. So it's really slippery road, which can bring you more troubles than fun.