Let's analyse each option:

1. The pH of neutral water at $298 \text{ K}$ is $7$, which is not zero. So, this statement is correct.
2. $\text{CH}_3\text{COOH}$ is a weak acid and $\text{NaOH}$ is a strong base. When they react in equimolar amounts ($1 \text{ M}$ each), they form sodium acetate ($\text{CH}_3\text{COONa}$), which is a salt of a weak acid and a strong base. Its aqueous solution undergoes anionic hydrolysis to produce a basic solution with $\text{pH} > 7$ . Thus, the statement that the pH will be $7$ is incorrect.
3. Dilute $\text{H}_2\text{SO}_4$ undergoes almost complete ionization, whereas concentrated $\text{H}_2\text{SO}_4$ contains largely unionized molecules. Furthermore, elevated temperatures increase the degree of dissociation. Hence, $[\text{H}^+]$ in dilute and hot $\text{H}_2\text{SO}_4$ is higher than in cold and concentrated $\text{H}_2\text{SO}_4$. So, this statement is correct.
4. A mixture of two acids ($\text{CH}_3\text{COOH}$ and $\text{HCl}$) will obviously be acidic, so its pH will be less than $7$. So, this statement is correct.

Therefore, the incorrect statement is Option 2.