How many games do you have to lose at 0 lp to get demoted?

if you just got in to new tier then 3-5 depending on your mmr if you've been there a while sometimes and you last 10 games or so mostly loses then as little as 3

just enough for your mmr to be lower than the division below you, or if your in a 5th division a whole league below you, ie you're gold5, you need to have bronze 1 mmr to fall into silver1.

RazorTS if you just got in to new tier then 3-5 depending on your mmr if you've been there a while sometimes and you last 10 games or so mostly loses then as little as 3

This unless your in level 5 of a division. Then you have to have the mmr of the level 5 below you. Eample

your silver 5 , to fall out of s5 you would need mmr of b5.