Why does Chi-Square Test need to meet the mutual exclusive assumption? In this page, it says Chi-Square must meet some assumption.

Assumption 2: All observations are independent.
It’s assumed that every observation in the dataset is independent.
That is, the value of one observation in the dataset does not affect
the value of any other observation.
Assumption 3: Cells in the contingency table are mutually exclusive.
It’s assumed that individuals can only belong to one cell in the
contingency table. That is, cells in the table are mutually exclusive
– an individual cannot belong to more than one cell.

Why does Chi-Square Test need to meet the mutual exclusive assumption? Is mutually exclusive assumption included in independent assumption?
What will happen when data do not meet that assumption?
Can anyone provide an example to explain it?

Any hypothesis test relies on correctly computing the null distribution -- the sampling distribution of the test statistic when $H_0$ is true. Any such calculation requires some set of assumptions -- some form of model which is intended to suit some specific circumstances, and on which basis the null distribution can be computed.
To derive an (asymptotic) chi-squared distribution for the test statistic (under $H_0$) you need to make some assumptions about how the counts arise. If those assumptions don't hold you won't get that distribution when $H_0$ is true, and so the critical values and p-values won't be correct (your test won't have the desired type I error rate).
Having an observation count in multiple categories is one of these situations; if you count each category it occurs in as a new count, you will miscalculate the variance of the test statistic, which relies on the fact that counts don't occur "together" like that; instead of having the usual negative dependence between category counts with fixed margins, there will be some (unaccounted for) positive dependence as well, and which may be relatively small but perhaps may be quite large. The precise effect depends on how these multiple-counts arise (specifically, their own dependence) and what proportion they are of the total.

Is mutually exclusive assumption included in independent assumption?

No. Firstly they're looking at different things (in the sense that the mutual exclusivity requirement is looking at counts for one sampling unit; a single person or whatever only contributes to a count in one level of any factor variable), and secondly counts for mutually exclusive events are negatively dependent, not independent.
