If you don't need dendrograms at the sides of the heatmap, the following are possible options:
Option 1. Create ggplot object & convert using plotly::ggplotly
:
library(ggplot2)
p <- ggplot(df,
aes(x = rankA, y = rankB, fill = V1, label = V2)) +
geom_tile() +
geom_label(fill = "white") +
viridis::scale_fill_viridis() # to match other packages' default palettes
plotly::ggplotly(p)

Option 2. Create plotly object:
plotly::plot_ly(
data = df,
x = ~rankA, y = ~rankB, z = ~V1, text = ~paste('V2: ', V2),
hoverinfo = "text",
type = "heatmap"
)
(This version shows "V2: /value of V2/" on hover. But it's not captured by the exported screenshot.)

If you really need the dendrograms, I'm afraid I haven't found a way to feed that into heatmaply's arguments. But times are desperate, you can consider the following...
Option 3. Create plotly object using heatmaply library & change the underlying code
# create the heatmap object hm (on hover, it will show the rankA / rankB / V1 values)
hm <- heatmaply::heatmaply(long_data = df %>%
select(rankB, rankA, V1) %>%
rename(name = rankB, variable = rankA, value = V1))
# look through hm's structure for hover text. I found it below, but I haven't
# used plotly enough to know if it's always going to be in the same place
str(hm)
> hm$x$data[4][[1]]$text
[,1] [,2]
[1,] "value: 8.636042<br />column: 0-1 w<br />row: 1-2 t" "value: 7.677400<br />column: 2-3 w<br />row: 1-2 t"
[2,] "value: 6.495266<br />column: 0-1 w<br />row: 3-5 t" "value: 5.420535<br />column: 2-3 w<br />row: 3-5 t"
[3,] "value: 5.480639<br />column: 0-1 w<br />row: 6-10 t" "value: 4.499810<br />column: 2-3 w<br />row: 6-10 t"
[4,] "value: 4.897840<br />column: 0-1 w<br />row: +10 t" "value: 3.496508<br />column: 2-3 w<br />row: +10 t"
attr(,"apiSrc")
[1] TRUE
# replace with V2's values, making sure the rows & columns are matched correctly
df %>%
select(-V1) %>%
mutate(V2 = paste("V2:", V2)) %>%
tidyr::spread(rankA, V2) %>%
arrange(factor(rankB, levels = c("1-2 t", "3-5 t", "6-10 t", "+10 t"))) %>%
select(-rankB) %>% as.matrix() -> hm$x$data[4][[1]]$text
hm
(As before, this version shows "V2: /value of V2/" on hover. But it's not captured by the exported screenshot.)

Data:
df <- read.table(header = T, stringsAsFactors = F,
text = ' rankA rankB V1 V2
1 "0-1 w" "1-2 t" 8.636042 10.43002
2 "0-1 w" "3-5 t" 6.495266 10.52126
3 "0-1 w" "6-10 t" 5.480639 10.56230
4 "0-1 w" "+10 t" 4.897840 10.64759
5 "2-3 w" "1-2 t" 7.677400 10.45409
6 "2-3 w" "3-5 t" 5.420535 10.47965
7 "2-3 w" "6-10 t" 4.499810 10.51640
8 "2-3 w" "+10 t" 3.496508 10.44883')