No, Tibetan is tonal.
Six tones in all: short high, long high, short low, long low, high falling, and low falling. So you could also say it has 4 tonal contrasts when we ignore rhyme duration: High, Low, High-fall, Low-fall. Very similar to Shanghainese (which has 2 tonal contrasts: high and low; with the high sometimes falling).
Only the Amdo/Qinghai dialect of Tibetan can be called non-tonal, this is probably a recent development.
Sino-Tibetan didn't start out being tonal. Archaic Chinese was completely non-tonal. Tibetan developed its tones a bit later than Sinitic, and is now gradually losing tonality just like a few of the Wu dialects. There is a Wu dialect right outside of Ningbo that can be considered non-tonal, while Shanghainese is on the verge of losing its High and Low distinction as well. What causes them to lose their tones? A lot has to do with contact with other groups of people. Many of the pitches in Wu dialects are exactly reversed from one another. Like Suzhou's falling and rising patterns was opposite of neighboring Songjiang; the mix of population from these regions led to the development of Shanghainese tones, which cancelled both out. It is also likely that tones were originally created by the same process.
Thanks, I always thought Tibetan was not tonal, because its script doesn't have tone marks.