Navy can be warm or cool, as can blue. Some have more yellow tones in than others.
I would have said that navy is quite warm toned. If you click on the smaller picture, of a woman who I assume bought it, it looks lovely on her and she has very yellow undertones.
When working out if a colour is warm or cool, I tend to imagine it as paint, and then mentally drop a bit of yellow or royal blue paint in and imagine which changes the "feel" of the colour more.