I've been reading up on Raggi's Lamentations of the Flame Princess, and was considering introducing it to my players using one of the adventures but using 1e/OSRIC rules. Would this break the whole concept? Would using 0e/LL rules without the LOTFP modifications?
If using a 1e/OSRIC group with an LotFP adventure, no modifications should be needed. The challenges are often not connected to character stats. The adventures should work equally well with any traditional ruleset.
If using LotFP characters with 1e/OSRIC modules, since those are usually more combat-heavy I would make sure the LotFP characters are one level higher than what the module recommends. For example, if an OSRIC module states it's for adventurers levels 4 - 7, I'd make sure the LotFP group is level 5 - 8.
I would make the same recommendation if using Basic Fantasy, Labyrinth Lord, or Swords & Wizardry characters with a 1e/OSRIC module, because 1e characters are generally built tougher, both due to bigger dice used for hit points and because of greater benefits for higher ability scores and a greater chance of having those higher scores.
I think you're good, for two reasons. First, Raggi's adventures are much more about the ambiance and the mood than the rules. I've been planning on using Death Frost Doom in a 4e campaign sometime, in fact. Second, the conversion between Lamentations (or any other OD&D variant) and AD&D is pretty straightforward anyhow.
And, yeah, they're intended to be used with AD&D anyhow. From the LotFP product page:
LotFP's role-playing books are supplements and adventures compatible with existing First Edition, Original Edition, and "Basic Edition" fantasy role-playing games.
The scenarios have very deliberately been written without much in the way of rules content, and what there is has been given in generic enough terms that it can be used as is with most versions of The Game. You may have trouble with D&D4, but even that should work; I ran Death Frost Doom in Rogue Trader, to give you a wild example.
I believe that Jim Raggi is intending to maintain the same format, even with his own rpg now on the shelves, so you should have no trouble.