int[] data = new int[] {10, 3, 0, 8, 4, 0, 4, 2 };
            //int[] data = new int[] { 0, 0, 0 };

            int min = 0;

            if (data.Any(d => d != 0))
                min = data.Where(d => d != 0).Min();

Or

            int[] data = new int[] {10, 3, 0, 8, 4, 0, 4, 2 };
            //int[] data = new int[] { 0, 0, 0 };

            int min = 0;

            IEnumerable<int> nonZero = data.Where(d => d != 0);
            if (nonZero.Any())
                min = nonZero.Min();