How do I remove the gaps between the shaded areas?

While I like the other answer, if you are fine with colors "blending" at the edges, we can simply use geomribbon() with group = 1 to create a continuous area.

library(ggplot2)
ggplot(taxodiversity, aes(x = intervalMa)) +
  geomribbon(aes(ymin = 0, ymax = speciesrichness, 
                  fill = period, group = 1), 
              alpha = 0.4) +
  geomline(aes(y = speciesrichness), 
            color = "black", group = 1) +
  labs(x = "Millions of years ago", y = "Number of species") +
  scalefillmanual(values = c("lightpink", "mediumpurple1", "skyblue", 
                               "darkolivegreen2", "orange")) +
  themeclassic() +
  theme(axis.text.x = elementtext(angle = 90, hjust = 1, vjust = 0.5),
        legend.position = "bottom")

taxodiversity 

I think the use of a factor on the x-axis is making this a little harder than it needs to be. I suggest changing that to using start itself as a continuous variable. With this, we can just grab the lead/lag of start and speciesrichness and calculate the mid points.

Note: your data has factors for intervalMa but we don't have enough to recreate that. Frankly, the factors (and their ordering) is not used here, I'll replicate what I think is happening by reversing the x axis.

Using the apparent data and your plot code (while reversing the x-axis), I see the following.

My change:

library(dplyr)
taxodiversity |>
  arrange(start, end) |>
  mutate(
    across(c(start, speciesrichness), ~ lag(.x), .names = "prev{.col}"),
    across(c(start, speciesrichness), ~ lead(.x), .names = "next{.col}")
  ) |>
  reframe(
    .by = period,
    start = c(first(prevstart + (start - prevstart)/2),
              last(start + (nextstart - start)/2) - 1e-9),
    speciesrichness = c(first(prevspeciesrichness + (speciesrichness - prevspeciesrichness)/2),
                         last(speciesrichness + (nextspeciesrichness - speciesrichness)/2))
  ) |>
  bindrows(taxodiversity) |>
  filter(!is.na(start), !is.na(speciesrichness)) |>
  arrange(start) |>
  ggplot(aes(x = start, y = speciesrichness)) +
  geomarea(aes(fill = period, group = period), alpha = 0.4) +
  geomline(color = "black", group = 1) +
  themeclassic() +
  labs(x = "Millions of years ago", y = "Number of species") +
  scalexreverse(breaks = taxodiversity$start, labels = taxodiversity$intervalMa) +
  scalefillmanual(values = c("lightpink", "mediumpurple1", "skyblue", "darkolivegreen2", "orange")) +
  theme(
    axis.text.x = elementtext(angle = 90, hjust = 1, vjust = 0.5),
    legend.position = "bottom"
  )

taxodiversity