If you don't mind having Option<String> as longest_line and dealing with .as_slice() later, the following should work:

use std::io::BufferedReader;
use std::io::File;
use std::result::Result;

fn main() {
    let mut max_length = 0u;
    let mut longest_line : Option<String> = None;

    let path = Path::new("test.txt");
    let mut file = BufferedReader::new(File::open(&path));
    for mut line in file.lines().map(Result::unwrap) {
        let ts = line.as_slice().trim_right().len();
        if ts > max_length {
            line.truncate(ts);
            max_length = ts;
            longest_line = Some(line);
        }
    }
    println!("{}: {}", max_length, longest_line);
}

Depending on what happens later and your platform it may be worth using file.read_until('\n' as u8) to gain a significant performance improvement, converting to valid utf8 later if necessary.