How do zeroes impact regression estimates?

Yes, it will definitely have an impact on your model. Here's a simple simulation to demonstrate. Feel free to fiddle with this code on your own.

It will change slope, intercept, and MSE. It could give you the illusion of a better fit because those (0,0) points will have very low residuals. If the center of your data is not (0, 0), then they'll be high-leverage too.

You need to consider the origin of your zeros. Are they true zeros? Or are they censored values? What to do depends on your question of interest.

library(ggplot2)
library(gridExtra)

# Specify model parameters
n_nonzero = 20
n_zero = 20

beta0 = 5
beta1 = 0.2

mean_x = 5


# Generate data without zeros
x = rnorm(n_nonzero, mean_x, 1)
y = beta0 + beta1 * x + rnorm(n_nonzero, 0, 1)

dat_no_zeros = data.frame(cbind(y, x))
dat_no_zeros = dat_no_zeros[sample(nrow(dat_no_zeros)),]
# Plot
p_no_zeros = ggplot(aes(x=x, y=y), data=dat_no_zeros) + 
  geom_point() + 
  geom_smooth(method='lm') +
  ggtitle('Without zeros')

# Add zeros to above data
x2 = c(x, rep(0, n_zero))
y2 = c(y, rep(0, n_zero))

dat_with_zeros = data.frame(cbind(y2, x2))
dat_with_zeros = dat_with_zeros[sample(nrow(dat_with_zeros)),]
# Plot
p_with_zeros = ggplot(aes(x=x2, y=y2), data=dat_with_zeros) + 
  geom_point() + 
  geom_smooth(method='lm') +
  ggtitle('With zeros')

grid.arrange(p_no_zeros, p_with_zeros, ncol=2)