Sort of, depending on how one understands the term ‘toxins’. It’s probably a phrase best avoided if one wishes to be scientifically credible – it may be better to say ‘yoga helps to stimulate the lymphatic system’. Deep breathing stimulates the lymphatic system. The body relies on the lymph system to remove foreign substances of biological origin, like the endotoxins and exotoxins produced by the turnover of microbes in the intestines (which live there all the time in normal, healthy people). The lymph vessels in the gut all drain ultimately into the thoracic duct; the movement of the lymph there is stimulated by breathing, through movements of the diaphragm. (While blood flowing in the arteries is pumped by the heart, both lymph and blood in the veins rely on muscle contraction to move forward.) Other poses which compress the abdomen, like Matsyendrasana, likely also assist in pushing lymph through the lymph system. (The largest lymph organ in the body is located in the abdomen – the gut-associated lymphoid tissue).
Most synthetic foreign substances which you eat (e.g. pharmaceuticals, food coloring, drugs of abuse, pesticides, etc) do not go through the lymph system. Rather they are absorbed into the bloodstream, chemically modified by the liver, continue on through the bloodstream, until they are excreted through the kidneys.
Some waste products from the actions of cells in the body are flushed by the lymph system. However, this is not true for the most well-known one – lactic acid, the substance responsible for the burning sensation in your muscles when you hold a pose for a long time. Once the muscular activity is over, oxygen arriving via the bloodstream will convert the lactic acid to pyruvic acid, which the cells can use.
Note that what constitutes a toxin may depend on the person. While we may happily eat bread, it is deadly poisonous for someone with celiac disease.
Finally, one should note that, in principle, all substances are toxic. “The dose makes the poison“, meaning that at a sufficiently high concentration anything can produce poisonous effects on the body.