I don't know too much about the specifics of 1e, but most of the differences between the two editions are small minor things that probably can be glossed over.
KaiiLurker's points are probably mostly right from what I know of 1e's differences. Module stuff, that is adventures, magic items and the like should be pretty compatible. Spells shouldn't be too hard to port. Monsters should be more or less useable, 2e stats are more detailed, but they tend to be things like fluff and non-combat information. Also monster XP is calculated differently in 2e so keep that in mind.
Classes shouldn't be a problem, but core 2e doesn't have any classes not present in 1e, 2e core has less classes than 1e core. The only class that has significant differences is the bard, but I've read some 1e groups like using the 2e bard because it's easier. There is the psionicist, but that's from a splat and psionics are optional anyway. Any other classes you want to use from optional 2e sources shouldn't be too much of a problem.
The one main subsystem in 2e that is probably different is proficiencies. BTB, it's optional so you can ignore it if you'd like anyway, but 2e's proficiency rules are supposed to be different from the rules that appeared in later 1e sources. There's also kits, but they're optional material from splats subject to DM approval. They're usually intended to be used with the proficiency rules as well. In any case, I don't see why proficiencies and kits couldn't just be ported to 1e largely whole anyway.
Then there's the various campaign settings which may or may not have problems. Here's what little I can offer on the settings (these are only general assumptions since I never heavily played any of the settings):
Greyhawk: 2e material will only matter here if you want to use stuff from Greyhawk Wars and beyond. Otherwise, the important setting material is 1e.
Dragonlance: I don't think there's any significant 2e material that would have any incompatibility with 1e. The Tales of the Lance box should be compatible with 1e, I don't know if there are issues with the Taladas material.
Forgotten Realms: Shouldn't be any problems. Al-Qadim falls under the Realms, main thing with that setting is that kits are suposed to be used (though I'd treat the sha'ir as a seperate class)
Mystara: Shouldn't be any problems. A good deal of fluff here comes from BECMI actually, the 2e material tends to be geared toward novice players so it's doubtful it would be a problem for a 1e group. There's also the Red Steel/Savage Coast stuff, main issue here is the Red Curse, but that probably shouldn't be a big problem with 1e.
Dark Sun: Requires 2e psionics rules. There's also stuff like defilers to take into account.
Ravenloft: I'm unaware of any issues that this setting might have with 1e.
Spelljammer: I'm unaware of any issues that this setting might have with 1e.
Planescape: Shouldn't be any problems. Planescape can support some rather diverse parties, so I'm sure it can probably handle 1e differences.
Birthright: The bloodline rules are a subsystem meant to be added on top of standard 2e classes, but I don't think using it under 1e rules might be too problematic. Domain rules are another subsystem, but I don't think it will clash with anything under 1e.