Solution: At least 11 chess pieces must be taken out to possibly meet the requirement. The reason is as follows: If a square is in the $i$-th row and the $j$-th column, then this square is denoted as $(i, j)$.

The first step is to prove that if any 10 chess pieces are taken, then among the remaining pieces, there must be a five-in-a-row, i.e., five pieces are consecutively aligned in a straight line (horizontally, vertically, or diagonally). Proof by contradiction. Assume that 10 chess pieces can be taken such that among the remaining pieces, there is no five-in-a-row. As shown in Figure 1, in the first five squares of each row, one piece must be taken, and in the first five squares of the last three columns, one piece must also be taken. Thus, the 10 taken pieces will not be distributed in the shaded area at the bottom right corner. By symmetry, they will not be distributed in the shaded areas at the other corners either. In the 1st and 2nd rows, one piece must be taken from each row, and they can only be placed in $(1,4),(1,5)$, $(2,4),(2,5)$. Similarly, at least two pieces must be taken from $(6,4),(6,5),(7,4),(7,5)$. In the 1st, 2nd, and 3rd columns, one piece must be taken from each column, distributed in the area of $(3,1),(3,2),(3,3),(4,1),(4,2),(4,3)$, $(5,1),(5,2),(5,3)$. Similarly, at least three pieces must be taken from the area of $(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)$. Thus, at least 10 pieces have already been taken in these areas. Therefore, no pieces can be taken from the central shaded area. Since at most 2 of the 4 pieces (1), (2), (3), (4) can be taken, there must be a five-in-a-row from the diagonal direction. Contradiction.

The second step is to construct a method to take 11 chess pieces, ensuring that among the remaining pieces, there is no five-in-a-row. As shown in Figure 2, as long as the pieces in the marked positions are taken, the remaining pieces will not form a five-in-a-row.

In summary, at least 11 chess pieces must be taken to possibly ensure that among the remaining pieces, there is no five-in-a-row.
Example 9 (2007 High School Mathematics League Additional Test Question) As shown in the figure, in a $7 \times 8$ rectangular chessboard, a chess piece is placed at the center of each small square. If two chess pieces are in small squares that share an edge or a vertex, then these two chess pieces are said to be connected. Now, some of the 56 chess pieces are removed so that no five remaining chess pieces are consecutively aligned in a straight line (horizontally, vertically, or diagonally). What is the minimum number of chess pieces that must be removed to meet this requirement? Explain your reasoning.