This should work, although I'm not familiarized with MySQL sql flavor.
Also, just to be sure, Is the class of this column in your database equivalent to POSIXct right?
filter_date <- as.POSIXct("2019-01-23", tz = "UTC")
tbl(con, "table_name") %>%
filter(created_at == filter_date) %>%
show_query()
# <SQL>
# SELECT *
# FROM `table_name`
# WHERE (`created_at` = '2019-01-23T00:00:00Z')