Use LINQ to get items in one List<>, that are not in another List<>

This can be addressed using the following LINQ expression:

var result = peopleList2.Where(p => !peopleList1.Any(p2 => p2.ID == p.ID));

An alternate way of expressing this via LINQ, which some developers find more readable:

var result = peopleList2.Where(p => peopleList1.All(p2 => p2.ID != p.ID));

Warning: As noted in the comments, these approaches mandate an O(n*m) operation. That may be fine, but could introduce performance issues, and especially if the data set is quite large. If this doesn't satisfy your performance requirements, you may need to evaluate other options. Since the stated requirement is for a solution in LINQ, however, those options aren't explored here. As always, evaluate any approach against the performance requirements your project might have.

If you override the equality of People then you can also use:

peopleList2.Except(peopleList1)

Except should be significantly faster than the Where(...Any) variant since it can put the second list into a hashtable. Where(...Any) has a runtime of O(peopleList1.Count * peopleList2.Count) whereas variants based on HashSet<T> (almost) have a runtime of O(peopleList1.Count + peopleList2.Count).

Except implicitly removes duplicates. That shouldn't affect your case, but might be an issue for similar cases.

Or if you want fast code but don't want to override the equality:

var excludedIDs = new HashSet<int>(peopleList1.Select(p => p.ID));
var result = peopleList2.Where(p => !excludedIDs.Contains(p.ID));

This variant does not remove duplicates.

Or if you want it without negation:

var result = peopleList2.Where(p => peopleList1.All(p2 => p2.ID != p.ID));

Basically it says get all from peopleList2 where all ids in peopleList1 are different from id in peoplesList2.

Just a little bit different approach from the accepted answer :)