Hi, I see now what happened. Many of your variables are factors, and many of these factors have very high number of levels, so even with few rows, a lot of information was needing to be passed to dput()
I wonder, whether the choice of factor vs character string, was explicitly yours in every case? it might be worth thinking about ?
anyway, before I look at your posted issue, I will make some amendments to your data to make it copy pasteable for others here that want to see with less effort.
using tidyverse subpackage(forcats) I was able to reduce the data like so (I named the dataset you provided to me as naughty_df
for obvious reasons 
df_to_share <- mutate_if(naughty_df,is.factor,forcats::fct_drop)
pryr::object_size(naughty_df)
#161 kB
pryr::object_size(df_to_share)
#8.7 kB
so the size was reduced by 20 times !
dput(df_to_share) is here:
df_to_share <- structure(list(id_cp_interaction = structure(c(2L, 1L, 5L, 4L,
3L), .Label = c("CPI-SS008FB916C", "CPI-SS03722104E", "CPI-SS07A03C20F",
"CPI-SS0D4CA5D92", "CPI-SS0D9366D11"), class = "factor"), interacting_pair = structure(c(5L,
4L, 1L, 2L, 3L), .Label = c("FGF1_FGFR4", "FGF18_FGFR4", "FGF19_FGFR4",
"FGF2_FGFR4", "HGF_CD44"), class = "factor"), partner_a = structure(c(5L,
4L, 3L, 1L, 2L), .Label = c("simple:O76093", "simple:O95750",
"simple:P05230", "simple:P09038", "simple:P14210"), class = "factor"),
partner_b = structure(c(1L, 2L, 2L, 2L, 2L), .Label = c("simple:P16070",
"simple:P22455"), class = "factor"), gene_a = structure(c(1L,
3L, 2L, 4L, 5L), .Label = c("ENSG00000019991", "ENSG00000113578",
"ENSG00000138685", "ENSG00000156427", "ENSG00000162344"), class = "factor"),
gene_b = structure(c(1L, 2L, 2L, 2L, 2L), .Label = c("ENSG00000026508",
"ENSG00000160867"), class = "factor"), secreted = structure(c(1L,
1L, 1L, 1L, 1L), .Label = "True", class = "factor"), receptor_a = structure(c(1L,
1L, 1L, 1L, 1L), .Label = "False", class = "factor"), receptor_b = structure(c(1L,
1L, 1L, 1L, 1L), .Label = "True", class = "factor"), annotation_strategy = structure(c(5L,
1L, 4L, 2L, 3L), .Label = c("curated", "I2D", "I2D,IMEx,InnateDB-All",
"I2D,IMEx,InnateDB-All,IntAct", "IMEx,InnateDB-All,IntAct"
), class = "factor"), is_integrin = structure(c(1L, 1L, 1L,
1L, 1L), .Label = "False", class = "factor"), BC.BC = c(0,
0, 0, 0, 0), BC.Erm = c(0, 0.4, 0.4, 0.4, 0.4), BC.Erp = c(1,
0, 0, 0.4, 0.2), Erm.BC = c(0, 0.6, 1, 0, 0), Erp.BC = c(0.4,
1, 0.4, 1, 1), Erm.Erm = c(0, 0.6, 1, 1, 0.6), Erm.Erp = c(1,
0.6, 0.6, 0.6, 0.6), Erp.Erm = c(0, 1, 0.4, 1, 1), Erp.Erp = c(1,
0.6, 0.4, 1, 1)), class = "data.frame", row.names = c(NA,
-5L))