What is the effect of increasing the sample size in ANOVA? Assume that a one-way ANOVA is performed on 3-4 groups of the same size.
What is the effect of increasing the sample size (equally for all groups)?
 A: Since the essential computation is the F-test:
$$F=\frac{\text{variance between treatments}}{\text{variance within treatments}}=\large\frac{\frac{\text{Sum Sqs}_{\text{treatments}}}{\text{no. treatments}-1}}{\frac{\text{Sum Sqs}_{\text{errors}}}{\text{no. cases}-\text{no. treatments}}}$$,
increasing the number of cases will decrease the denominator, and increase the $\ F$ test statistic, making it more likely to obtain a small p-value with everything else constant.
In other words, it will result in increased power, and decreased type II errors.
And following up on the comments from @whuber, the increase in the number of observations also has an effect in the mean squared residuals (MSR or RSE), and by extension on the standard error (SE) on the estimates. This is clear when considering that the standard error of the estimates is simply the square root of $\widehat{\textrm{Var}}(\hat{\mathbf{\beta}}) = \hat{\sigma}^2  (\mathbf{X}^{\prime} \mathbf{X})^{-1}$, and that the model matrix $X$ in this case of one-way ANOVA is as simple as an intercept and dummy coded entries:

with the defining value $\hat\sigma^2=\frac{u^Tu}{\text{df}}$ and $\text{df}=\text{cases}-\text{groups}-1$, which is roughly the mean of the squared residuals. Carrying out a Monte Carlo simulation by drawing three groups of observations from normal distributions with the same variance $\sigma^2=9$, but with means $x_1=10$, $x_2=15$ and $x_3=20$ 

with increasing numbers of balanced observations from $5$ to $1000$ we can see how the mean squared of the residuals display a funneled shape, rapidly dropping the spread of the mean squared residuals as the sample size increases:

