If you want to just remove the psi1. with different numbers you can use str_replace:
df <- data.frame(var = c("psi2.1", "psi1.2", "psi33.55", "psi12.42"))
df %>% mutate(var = str_replace(var, "psi(\\d+)\\.", ""))
#   var
# 1   1
# 2   2
# 3  55
# 4  42
Solution by @Jaap:
gsub('psi\\d+\\.', '', psi2$variable)
And benchmark, I have added stringi::stri_replace_first_regex and perl = TRUE to gsub:
microbenchmark::microbenchmark(
  str_replace = str_replace(df$var, "psi\\d+\\.", ""),
  stri_replace_all = stringi::stri_replace_first_regex(df$var, "psi\\d+\\.", ""),
  sub = sub(".*\\.", "", df$var),
  gsub = gsub('psi\\d+\\.', '', df$var),
  gsub_perl = gsub('psi\\d+\\.', '', df$var, perl = TRUE),
  times = 10000
)
Unit: microseconds
             expr    min      lq      mean  median      uq       max neval
      str_replace 96.661 106.101 129.08727 110.632 117.805  3951.009 10000
 stri_replace_all 28.319  33.228  41.57426  36.626  39.647  1980.413 10000
              sub 14.349  17.369  22.21423  19.257  23.033  1682.124 10000
             gsub 18.879  22.278  34.89121  24.921  28.697 63495.163 10000
        gsub_perl 76.272  79.293  88.32751  81.558  84.956  1865.251 10000
The sub solution is the fastest.
Applying the benchmark on a larger dataset:
df <- df[sample(nrow(df), 1e6, replace = TRUE), , drop = FALSE]
microbenchmark::microbenchmark(
  str_replace = str_replace(df$var, "psi\\d+\\.", ""),
  stri_replace_all = stringi::stri_replace_first_regex(df$var, "psi\\d+\\.", ""),
  sub = sub(".*\\.", "", df$var),
  gsub = gsub('psi\\d+\\.', '', df$var),
  gsub_perl = gsub('psi\\d+\\.', '', df$var, perl = TRUE),
  times = 50
)
the result:
Unit: milliseconds
             expr      min       lq     mean   median       uq      max neval  cld
      str_replace 293.2773 301.9520 311.9032 308.5192 322.4974 344.7649    50  b  
 stri_replace_all 294.8729 298.8479 316.9213 306.4369 317.3555 518.5287    50  b  
              sub 468.2134 473.1803 487.0336 485.1354 498.1503 527.2476    50   c 
             gsub 649.6209 673.4312 690.7942 683.6022 701.3134 909.2599    50    d
        gsub_perl 251.0663 255.1404 263.9778 260.3426 274.6684 287.3492    50 a