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You have a function which is performing three steps:

1. Copy the input range
2. Sort the copied range by distance to a given element
3. Erase elements from the result range

I would omit the first and last step. These are convenience elements providing no real functionality. In addition, the third step is erasing possible useful information and involves undefined behavior if the input range has not the number of requested elements (result.erase(std::begin(result) + n, std::end(result);).

Leaving the second step: Here we have a std::sort with a custom comparator operating on distances. Your comparator depends on the result type of a distance function and is not more than a type trait. You might avoid that.

An alternative implementation might be:

#include <algorithm>
#include <iterator>

// Distance
// ========

template <typename T>
struct Distance {
    T operator () (const T& a, const T&  b) {
        return std::abs(a - b);
    }
};

// Compare Distance
// ================

template <
    typename T,
    typename DistanceFunctor = Distance<T>,
    typename CompareFunctor = std::less<decltype(
        std::declval<DistanceFunctor>()(std::declval<T>(), std::declval<T>()))>>
struct CompareDistance
{
    T pivot;
    DistanceFunctor distance;
    CompareFunctor compare;
    CompareDistance(T&& pivot)
    :   pivot(std::move(pivot))
    {}

    CompareDistance(T&& pivot, DistanceFunctor&& distance)
    :   pivot(std::move(pivot)),
        distance(std::move(distance))
    {}

    CompareDistance(T&& pivot, DistanceFunctor&& distance, CompareFunctor&& compare)
    :   pivot(std::move(pivot)),
        distance(std::move(distance)),
        compare(std::move(compare))
    {}

    bool operator () (const T& a, const T& b) {
        return compare(distance(a, pivot), distance(b,  pivot));
    }
};

// Distance Sort
// =============

template <typename Iterator, typename T>
inline void distance_sort(
    Iterator first,
    Iterator last,
    T&& pivot)
{
    typedef typename std::iterator_traits<Iterator>::value_type value_type;
    CompareDistance<value_type> compare_distance(std::move(pivot));
    std::sort(first, last, compare_distance);
}

template <typename Iterator, typename T, typename Distance>
inline void distance_sort(
    Iterator first,
    Iterator last,
    T&& pivot,
    Distance&& distance)
{
    typedef typename std::iterator_traits<Iterator>::value_type value_type;
    CompareDistance<value_type, Distance> compare_distance(
        std::move(pivot),
        std::move(distance));
    std::sort(first, last, compare_distance);
}

template <typename Iterator, typename T, typename Distance, typename Compare>
inline void distance_sort(
    Iterator first,
    Iterator last,
    T&& pivot,
    Distance&& distance,
    Compare&& compare)
{
    typedef typename std::iterator_traits<Iterator>::value_type value_type;
    CompareDistance<value_type, Distance, Compare> compare_distance(
        std::move(pivot),
        std::move(distance),
        std::move(compare));
    std::sort(first, last, compare_distance);
}

// Test
// ====

#include <iostream>

int main() {
    std::vector<int> original = { 56, 10, 79841651, 45, 59, 68, -20, 0, 36, 23, -3256 };

    // Find closest neighbours [less]:
    std::vector<int> elements(original);
    distance_sort(begin(elements), end(elements), 4);

    for(const auto& e : elements)
        std::cout << e << ' ';
    std::cout << '\n';

    // Find closest neighbours [greater]:
    distance_sort(begin(elements), end(elements), 4, Distance<int>(), std::greater<int>());

    for(const auto& e : elements)
        std::cout << e << ' ';
    std::cout << '\n';

    // Without distance_sort, but with existing tools
    std::sort(
        begin(elements),
        end(elements),
        [](int a, int b) {
            const int pivot = 4;
            return std::abs(a - pivot) < std::abs(b - pivot);
        }
    );

    for(const auto& e : elements)
        std::cout << e << ' ';
    std::cout << '\n';
}

Please notice the option not to provide anything and rely on existing tools.