Have the players actually read the adventure, or do they just have access to read it? And, ask them honestly, how well do they remember it?
If you can trust the maturity of your players (which hopefully you can), they won't go out of their way to crib up on the adventure before they start playing, nor will they go out of their way to spoil adventure elements for new players. It might not hurt to encourage the few who know the adventure to try and take a back-seat during strategy time, but hopefully they will know to do this on their own. (In other words, don't say, "I think we should go left because there is a big bugbear to the right and we can't handle that right now.")
On the other hand, I played through the opening dungeon of LMoP after reading it myself (with the intention of running it), and it was a little boring as a player. I couldn't really engage with the problem-solving, exploration aspects, because there was no actual discovery in it for me, so I just played dumb and impulsive, and hoped that the other players would play smart and inquisitive. The actual combat encounters were interesting, because you never know how a DM is going to make a group of NPCs come to life, but sneaking around the cave (normally one of favorite aspects of an adventure), was pretty dull.
If players are honest with you about which parts of the adventure they have read, consider:
a) swapping those parts out entirely. (replace the ambush and goblin cave from the beginning with a similar dungeon of your own design, or steal a similarly-sized dungeon from another adventure)
b) make a few significant changes to the existing adventure to keep new players guessing. (And let them know you've done this.) Move or swap out traps. Switch up where some of the encounters take place. Make some changes to the layout. Make just enough changes so that players feel like they have a reason to pay attention during the exploration aspects, and to keep some of the tension up.
I think it could work, and if you love the adventure and want to run it with a (mostly) new group, I'd say go for it. Just try to work a little to keep things fresh for your experienced folks so they don't feel like they have to tune out.