In my opinion if you teach younger children, such as elementary school students, its a big gigantic NO, I don’t believe it is right to have any out-of-school relationship with younger students unless its a side effect of being friends with their parents.
But if you teach high school or older students, I say go right ahead.
A teacher has to have the students’ respect if they truly want to have students succeed in their classroom. This is obviously very abstract, and every kid has their own subconscious rules about who they will respect and for different reasons. A lot of the teachers I respected in high school only received that respect because I thought they deserved it. Every teacher has their own way of gaining the majority’s respect. Some were intimidating dicks, some were funny, some would make sure the work they gave you wasn’t overbearing and cut out the fluff, etc. Keep in mind that teachers are always on the clock when it comes to dealing with their students. Do what you want, but remember that your job is to make sure those students succeed in your class.
Some common sense advice that you probably already know:
If any authority or parent says stop, just stop, with all of them.
Don’t curse, or yell when playing these games, its a hard one, but as a teacher you probably have the ability to suppress this kind of stuff.
If they didn’t finish their homework, refuse to play with them, and don’t play with them too late, whether or not there is school the next day.
Daddypants is right, you aren’t their friend, you are their teacher, make sure they know that you will treat them the same whether or not they play games with you.
And keep it casual, do not make any plans to play with them. Never tell them that you might “see them later/catch them online later” if you happen to be online, cool, if not then no biggie.