Read the data in lines (use fgets()).
If the line contains a #, terminate the string there by replacing the '#' with '\0'.  Then scan the line for numbers.
See also How to use sscanf() in loops?
And don't forget to check that the file was opened.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
int main(void)
{
    char filename[20];
    printf("Please enter the name of file to load: ");
    if (scanf("%19s", filename) != 1)
    {
        fprintf(stderr, "Failed to read file name\n");
        return 1;
    }
    FILE *lun = fopen(filename, "r");
    if (lun == NULL)
    {
        fprintf(stderr, "Failed to open file %s for reading\n", filename);
        return 1;
    }
    char line[4096];
    int min = 0;  // Avoid compilation warnings (may be used uninitialized)
    int max = 0;  // Ditto
    int sum = 0;
    int count = 0;
    while (fgets(line, sizeof(line), lun) != NULL)
    {
        char *hash = strchr(line, '#');
        if (hash != NULL)
            *hash = '\0';
        int pos;
        int num;
        int off = 0;
        while (sscanf(line + off, "%d%n", &num, &pos) == 1)
        {
            if (count == 0)
                min = max = num;
            if (num > max)
                max = num;
            if (num < min)
                min = num;
            sum += num;
            count++;
            off += pos;  // Skip through line
        }
    }
    fclose(lun);
    printf("Minimum value: %d\nMaximum value: %d\nAverage value: %lf\n",
              min, max, sum / (double)count);
    return 0;
}
If your compiler doesn't support C99 or later, you will have to move variable declarations to the start of a block (immediately after a {).
Handling doubles isn't really any harder:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
int main(void)
{
    char filename[20];
    printf("Please enter the name of file to load: ");
    if (scanf("%19s", filename) != 1)
    {
        fprintf(stderr, "Failed to read file name\n");
        return 1;
    }
    FILE *lun = fopen(filename, "r");
    if (lun == NULL)
    {
        fprintf(stderr, "Failed to open file %s for reading\n", filename);
        return 1;
    }
    char line[4096];
    double min = 0.0;   // Avoids 'used when uninitialized' warnings
    double max = 0.0;   // Avoids 'used when uninitialized' warnings
    double sum = 0;
    int count = 0;
    while (fgets(line, sizeof(line), lun) != NULL)
    {
        char *hash = strchr(line, '#');
        if (hash != NULL)
            *hash = '\0';
        int pos;
        double num;
        int off = 0;
        while (sscanf(line + off, "%lf%n", &num, &pos) == 1)
        {
            if (count == 0)
                min = max = num;
            if (num > max)
                max = num;
            if (num < min)
                min = num;
            sum += num;
            count++;
            off += pos;  // Skip through line
        }
    }
    fclose(lun);
    printf("Minimum value: %f\nMaximum value: %f\nAverage value: %f\n",
              min, max, sum / count);
    return 0;
}