Reshape data to the long form and filter dates prior to the forest fire.

library(tidyverse)

data1989 %>%
  pivot_longer(-date_fire, names_to = "date") %>%
  mutate(date_fire = as.Date(date_fire),
         date = as.Date(date, "X%Y.%m.%d")) %>%
  filter(date < date_fire) %>%
  group_by(date_fire) %>%
  summarise(meanPreFire = mean(value, na.rm = T))

# # A tibble: 3 x 2
#   date_fire  meanPreFire
#   <date>           <dbl>
# 1 1987-01-01       0.62 
# 2 1987-07-03       0.559
# 3 1988-01-01       0.764

We can use base R, by creating a row/column index. The column index can be got from findInterval with the column names and the 'date_fire'

j1 <- findInterval(as.Date(data1989$date_fire), as.Date(names(data1989)[-1]))
l1 <- lapply(j1+1, `:`, ncol(data1989)-1)   
m1 <- cbind(rep(seq_len(nrow(data1989)), j1), sequence(j1))
m2 <- cbind(rep(seq_len(nrow(data1989)), lengths(l1)), unlist(l1))
data1989$meanPreFire <- tapply(data1989[-1][m1], m1[,1], FUN = mean, na.rm = TRUE)
data1989$meanPostFire <- tapply(data1989[-1][m2], m2[,1], FUN = mean, na.rm = TRUE)


data1989
#   date_fire 1986-01-01 1986-06-03 1986-10-19 1987-01-19 1987-06-19 1987-10-19 1988-01-19 1988-06-19 1988-10-19
#1 1987-01-01      0.500      0.560       0.80       0.75      0.100       0.15       0.20       0.18       0.21
#2 1987-07-03      0.589      0.447         NA       0.65      0.550       0.12       0.22       0.21       0.24
#3 1988-01-01      0.660      0.750       0.83       0.75      0.811       0.78       0.32       0.23       0.25
#  meanPreFire meanPostFire
#1      0.6200    0.2650000
#2      0.5590    0.1975000
#3      0.7635    0.2666667

Or using melt/dcast from data.table

library(data.table)
dcast(melt(setDT(data1989), id.var = 'date_fire')[, 
    .(value = mean(value, na.rm = TRUE)), 
    .(date_fire, grp = c('postFire', 'preFire')[1 + (as.IDate(variable) < as.IDate(date_fire))]) ], date_fire ~ grp)[data1989, on = .(date_fire)]
#    date_fire  postFire preFire 1986-01-01 1986-06-03 1986-10-19 1987-01-19 1987-06-19 1987-10-19 1988-01-19 1988-06-19
#1: 1987-01-01 0.2650000  0.6200      0.500      0.560       0.80       0.75      0.100       0.15       0.20       0.18
#2: 1987-07-03 0.1975000  0.5590      0.589      0.447         NA       0.65      0.550       0.12       0.22       0.21
#3: 1988-01-01 0.2666667  0.7635      0.660      0.750       0.83       0.75      0.811       0.78       0.32       0.23
#   1988-10-19
#1:       0.21
#2:       0.24
#3:       0.25

data

data1989 <- data.frame("date_fire" = c("1987-01-01", "1987-07-03", "1988-01-01"), 
                       "1986-01-01" = c(0.5, 0.589, 0.66), 
                       "1986-06-03" = c(0.56, 0.447, 0.75), 
                       "1986-10-19" = c(0.8, NA, 0.83),
                       "1987-01-19" = c(0.75, 0.65,0.75), 
                       "1987-06-19" = c(0.1, 0.55,0.811),
                       "1987-10-19" = c(0.15, 0.12, 0.780),
                       "1988-01-19" = c(0.2, 0.22,0.32), 
                       "1988-06-19" = c(0.18, 0.21,0.23),
                       "1988-10-19" = c(0.21, 0.24, 0.250), check.names = FALSE,
                       stringsAsFactors = FALSE) 

The solution would be much more concise if we would keep the data in long(er) form... but this reproduces the desired output:

library(dplyr)
library(tidyr)
data1989 %>% 
  pivot_longer(-date_fire, names_to = "date_NDVI", values_to = "value", names_prefix = "^X") %>% 
  mutate(date_fire = as.Date(date_fire, "%Y-%m-%d"),
         date_NDVI = as.Date(date_NDVI, "%Y.%m.%d")) %>% 
  group_by(date_fire) %>% 
  mutate(period = ifelse(date_NDVI < date_fire, "before_fire", "after_fire")) %>% 
  group_by(date_fire, period) %>% 
  mutate(average_NDVI = mean(value, na.rm = TRUE)) %>% 
  pivot_wider(names_from = date_NDVI,  names_prefix = "X", values_from = value) %>% 
  pivot_wider(names_from = period, values_from = average_NDVI) %>% 
  group_by(date_fire) %>% 
  summarise_all(funs(sum(., na.rm=T)))

Returns:

# A tibble: 3 x 12
  date_fire  `X1986-01-01` `X1986-06-03` `X1986-10-19` `X1987-01-19` `X1987-06-19` `X1987-10-19` `X1988-01-19` `X1988-06-19` `X1988-10-19` before_fire after_fire
  <date>             <dbl>         <dbl>         <dbl>         <dbl>         <dbl>         <dbl>         <dbl>         <dbl>         <dbl>       <dbl>      <dbl>
1 1987-01-01         0.5           0.56           0.8           0.75         0.1            0.15          0.2           0.18          0.21       0.62       0.265
2 1987-07-03         0.589         0.447          0             0.65         0.55           0.12          0.22          0.21          0.24       0.559      0.198
3 1988-01-01         0.66          0.75           0.83          0.75         0.811          0.78          0.32          0.23          0.25       0.764      0.267

Edit:

If we stop the expression right after calculating the averages we can use the data in this structure to easily calculate the variance or account for variable number of observations. I think it's ok to keep the date_fireas its own column, but I'd suggest leaving the other dates as a column (because they correspond to observations). Especially if we want to do more analysis with the data using ggplot2 and other tidyverse functions.