How to ensure that text does not intersect with a ggplot graph?

An easy option to achieve your desired result is the "secondary axis trick", i.e. add the labels via a secondary x axis. This can be achieved by "adding"

scalexcontinuous(
  sec.axis = dupaxis(
    name = NULL,
    breaks = vlines$xintercept,
    labels = c("Intervention\nbegins", "Intervention\nends")
  )
)

And in case the left alignment of the labels is required I would suggest to switch to ggtext::elementmarkdown for the top axis text which provides an additional halign= parameter to achieve that (but requires to switch to "
" to add a line break):

scalexcontinuous(
  sec.axis = dupaxis(
    name = NULL,
    breaks = vlines$xintercept,
    labels = c("Intervention
begins", "Intervention
ends")
  )
) +
theme(
  axis.text.x.top = ggtext::elementmarkdown(
    halign = 0
  )
)

Full reproducible code for the second case:

library(ggplot2)
ggplot(df, aes(x = time, y = value)) +
  geomrect(
    data = shadedregion,
    aes(xmin = xmin, xmax = xmax, ymin = ymin, ymax = ymax, fill = Period),
    inherit.aes = FALSE, alpha = 0.3
  ) +
  geomline(aes(color = "Observed Data"), linewidth = 0.8) +
  geomvline(
    data = vlines,
    aes(xintercept = xintercept, linetype = Line),
    linewidth = 1
  ) +
  scalefillmanual(name = NULL, values = c("Intervention Period" = "lightgreen")) +
  scalecolormanual(name = NULL, values = c("Observed Data" = "steelblue")) +
  scalelinetypemanual(name = NULL, values = c("Intervention Start/End" = "dotted")) +
  labs(x = "Time", y = "Value", title = "Time Series with Intervention Period") +
  thememinimal() +
  theme(
    legend.position = "bottom"
  ) +
  scalexcontinuous(
    sec.axis = dupaxis(
      name = NULL,
      breaks = vlines$xintercept,
      labels = c("Intervention
begins", "Intervention
ends")
    )
  ) +
  theme(
    axis.text.x.top = ggtext::elementmarkdown(
      halign = 0
    )
  ) +
  guides(
    fill = guidelegend(order = 1),
    color = guidelegend(order = 2),
    linetype = guidelegend(order = 3)
  )

One way is to make another plot containing just the title and the labels, and then combine the two using, say, the plotgrid function from cowplot

datalabs