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Pasting your code exactly as is into my IDE, ReSharper gives the following warnings and suggestions:

List<List<int>> returnResult = new List<List<int>>();

Use implicitly typed local variable declaration

I'd also rename returnResult to just result, so:

var result = new List<List<int>>();

var newList = new List<int>();

newList.Add(number);

Use collection initializer

Each item in a collection initializer calls Add on the list, so this is equivalent:

var newList = new List<int> {number};

listMinusOneObject.Remove(listMinusOneObject.Where(x => x == number).First());

Replace with single call to First(..)

I'd ignore that suggestion, as this SO question/answer demonstrates a performance improvement with .Where().First() over .First().

if (listMinusOneObject.Count() > 0)

Use method Any()

That I'd do. Count() will iterate every item in the list, only to compare against 0; Any() will return true as soon as it finds 1 item in the list:

if (listMinusOneObject.Any())

When a foreach loop can be converted to a LINQ statement, ReSharper is very good at pointing it out. So I replaced your for loops with foreach loops (removed the number declaration/assignment):

var distinctInts = ints.Distinct().ToList();
foreach(var number in distinctInts)
{
 var newList = new List<int> {number};
 returnResult.Add(newList);
 var listMinusOneObject = ints.Select(x => x).ToList();
 listMinusOneObject.Remove(listMinusOneObject.Where(x => x == number).First());
 if (listMinusOneObject.Any())
 {
 _GetAllCombinationsOfAllSizes(listMinusOneObject, newList, ref returnResult);
 }
}

and

var distinctInts = ints.Distinct().ToList();
foreach (var number in distinctInts)
{
 var newList = growingList.ToList();
 newList.Add(number);
 returnResult.Add(newList);
 var listMinusOneObject = ints.Select(x => x).ToList();
 listMinusOneObject.Remove(listMinusOneObject.Where(x => x == number).First());
 if (listMinusOneObject.Any())
 {
 _GetAllCombinationsOfAllSizes(listMinusOneObject, newList, ref returnResult);
 }
}

And as I expected, ReSharper doesn't suggest anything. Why is that so?

LINQ stands for Language-INtegrated Query. Querying objects does not usually involve side-effects, and this is precisely what's going on in your loops' bodies: you're not querying a List<int>, you're iterating it. A foreach loop does that.

That was just cosmetics and minor adjustments (some of which may have a positive impact on performance); there's a lot to be said about your algorithm, I'll leave that to another reviewer.