In order for spread to work as intended, the resulting data frame must have uniquely identified rows and columns. In the case of your data, the "date" column is the only unique identifier after spreading. However, rows 36 and 38 are identical:
date tmin state
36 2018-01-03 -3 OH
38 2018-01-03 -3 OH
This puts tidyr in the impossible position of trying to resolve two data points to the same row and column. In addition, rows 35 and 37 both have the same date and state, once again creating the impossible situation of placing two different values into the same position in the new data frame:
date tmin state
35 2018-01-03 NA UT
37 2018-01-03 22 UT
The following data cleanup will make spreading possible:
df %>%
filter(!is.na(tmin)) %>% # remove NA values
unique %>% # remove duplicated rows
spread(state, tmin)
date OH UT
1 2018-01-02 -4 24
2 2018-01-03 -3 22
3 2018-01-04 11 19
4 2018-01-05 3 23
5 2018-01-06 0 29
...
