Generic IEqualityComparer<T> and GetHashCode

Nothing would go wrong, but in hash-table based containers, you're going from approx O(1) to O(n) performance when doing a lookup. You'd be better off simply storing everything in a List and brute force searching it for items that fulfil equality.

If a common use-case is comparing objects according to one of their properties, you could add an additional constructor and implement and call it like this:

public GenericEqualityComparer(Func<T, object> projection)
{
    compareFunction = (t1, t2) => projection(t1).Equals(projection(t2));
    hashFunction = t => projection(t).GetHashCode();
}

var comaparer = new GenericEqualityComparer( o => o.PropertyToCompare);

This will automatically use the hash as implemented by the property.

EDIT: a more efficient and robust implementation inspired my Marc's comments:

public static GenericEqualityComparer<T> Create<TValue>(Func<T, TValue> projection)
{
    return new GenericEqualityComparer<T>(
        (t1, t2) => EqualityComparer<TValue>.Default.Equals( projection(t1), projection(t2)),
        t => EqualityComparer<TValue>.Default.GetHashCode(projection(t)));
}

var comparer = GenericEqualityComparer<YourObjectType>.Create( o => o.PropertyToCompare);