Visualising a three way interaction between two continuous variables and one categorical variable in R

Here are a couple of options for visualizing the model output in two dimensions. I'm assuming here that the goal here is to compare Treatment to Control

library(tidyverse)
  theme_set(theme_classic() +
          theme(panel.background=element_rect(colour="grey40", fill=NA))

dat = read_excel("Some Data.xlsx")  # I downloaded your data file

mod <- lm(DV ~ IVContinuousA * IVContinuousB * IVCategorical, data=dat)

# Function to create prediction grid data frame
make_pred_dat = function(data=dat, nA=20, nB=5) {
  nCat = length(unique(data$IVCategorical))
  d = with(data, 
           data.frame(IVContinuousA=rep(seq(min(IVContinuousA), max(IVContinuousA), length=nA), nB*2),
                      IVContinuousB=rep(rep(seq(min(IVContinuousB), max(IVContinuousB), length=nB), each=nA), nCat),
                      IVCategorical=rep(unique(IVCategorical), each=nA*nB)))

  d$DV = predict(mod, newdata=d)

  return(d)
}

IVContinuousA vs. DV by levels of IVContinuousB

The roles of IVContinuousA and IVContinuousB can of course be switched here.

ggplot(make_pred_dat(), aes(x=IVContinuousA, y=DV, colour=IVCategorical)) + 
  geom_line() +
  facet_grid(. ~ round(IVContinuousB,2)) +
  ggtitle("IVContinuousA vs. DV, by Level of IVContinousB") +
  labs(colour="")

You can make a similar plot without faceting, but it gets difficult to interpret as the number of IVContinuousB levels increases:

ggplot(make_pred_dat(nB=3), 
       aes(x=IVContinuousA, y=DV, colour=IVCategorical, linetype=factor(round(IVContinuousB,2)))) + 
  geom_line() +
  #facet_grid(. ~ round(IVContinuousB,2)) +
  ggtitle("IVContinuousA vs. DV, by Level of IVContinousB") +
  labs(colour="", linetype="IVContinuousB") +
  scale_linetype_manual(values=c("1434","11","62")) +
  guides(linetype=guide_legend(reverse=TRUE))

Heat map of the model-predicted difference, DV treatment - DV control on a grid of IVContinuousA and IVContinuousB values

Below, we look at the difference between treatment and control at each pair of IVContinuousA and IVContinuousB.

ggplot(make_pred_dat(nA=100, nB=100) %>% 
         group_by(IVContinuousA, IVContinuousB) %>% 
         arrange(IVCategorical) %>% 
         summarise(DV = diff(DV)), 
       aes(x=IVContinuousA, y=IVContinuousB)) + 
  geom_tile(aes(fill=DV)) +
  scale_fill_gradient2(low="red", mid="white", high="blue") +
  labs(fill=expression(Delta*DV~(Treatment - Control)))

If you really want to avoid 3-d plotting, you could indeed turn one of the continuous variables into a categorical one for visualization purposes.

For the purpose of the answer, I used the Duncan data set from the package car, as it is of the same form as the one you described.

library(car)
# the data
data("Duncan")

# the fitted model; education and income are continuous, type is categorical
lm0 <- lm(prestige ~ education * income * type, data = Duncan)

# turning education into high and low values (you can extend this to more 
# levels)
edu_high <- mean(Duncan$education)  + sd(Duncan$education)
edu_low <- mean(Duncan$education)  - sd(Duncan$education)

# the values below should be used for predictions, each combination of the 
# categories must be represented:
prediction_mat <- data.frame(income = Duncan$income, 
                         education = rep(c(edu_high, edu_low),each = 
                         nrow(Duncan)),
                         type = rep(levels(Duncan$type), each = 
                         nrow(Duncan)*2))


predicted <- predict(lm0, newdata = prediction_mat)


# rearranging the fitted values and the values used for predictions
df <- data.frame(predicted,
             income = Duncan$income,
             edu_group =rep(c("edu_high", "edu_low"),each = nrow(Duncan)),
             type = rep(levels(Duncan$type), each = nrow(Duncan)*2))


# plotting the fitted regression lines
ggplot(df, aes(x = income, y = predicted, group = type, col = type)) + 
geom_line() + 
facet_grid(. ~ edu_group)