I was thinking of running LMOP with an online group, and I need some help.

First, are there any races/classes I shouldn't allow because it would break the module?

Second, are there any good modifications to make so that it may seem like a new adventure?

And third, I'm currently playing this module. Should I finish my game before doing this?(because spoilers)

Yes, finish playing through the adventure before you run it. Also read it all the way through.
Off hand, I can't think of any races/classes or combination that would be a problem.  It is low level and pretty well balanced from what I recall.
As for making it seem new... you might want to consider incorporating some of the material from the Essentials Kit.  This is a second set of low level encounters centered in and around Phandalin.  There are also numerous modules on the Dungeon Master's Guild set in the same area and based on the original LMoP content.

There shouldn't be a problem with any races and specific classes. I say it would be whatever you would feel comfortable with, if you want to limit some things. 

You can also consider mixing in Dragón of Icespire Peak and even In Volo's Wake, since all takes place around Phandalin. 

 Third, it's up to you. I mean if you feel that you'll play differently because you'll know about what's coming up, then maybe avoid it. 

Thanks.