Sometime は is used when you making a contrast, and when you specify something that you don't eat it's always a contrast since you presumably -do- eat other foods, and it's especially a contrast when you've just named some foods that you do eat.
You'll find that は can replace が or を to make a change in emphasis like this.
You can say にくをたべません in some other contexts, if you were comparing what you eat to what someone else eats, or comparing what you're having for lunch and what you're having for dinner. But when you're making the general statement 'I don't eat meat', then you use にくはたべません。