If Everyone Were to Go Vegan/Vegetarian....

Theoretically speaking.

 Animals will be liberated (yay!)

Number of animals will increase sharply. Including herbivores and omnivores. 

Humans and animals both eat plant products. 

Increased competition for plant and plant products between humans and animals. 

Increased demand for food equals lesser food for the animals. 

Which will then lead to a lack of food for the animals.  

A lack of food for the animals means animals starve and go through starvation.

Animals then die due to starvation and a lack of food.

Animals will still die nevertheless.