I heard it's like a month max before you start slowly losing LP.
If you're already low LP, you start losing divisions even more slowly.
Here's the excert from the FAQ in the client:
What happens if I stop playing ranked for awhile?
After 28 days of inactivity in standard tiers a couple things will happen. First, you'll be hidden from the standings in your League, and you may lose League Points depending on your current tier. Every seven days thereafter, you'll lose LP again until you play a match in that queue. In apex tiers, to ensure only the top active players are being showcased the number of days before becoming inactive is 10. LP losses will be incurred each day of inactivity after.
The number of LP lost following each period of decay is as follows:
Master & Challenger: 250
Diamond: 50
Platinum: 35
If you fall below zero LP as a result of decay, you'll be placed into the next lowest division. If you're already in Division V of your league, you'll fall into a new league in the next tier down.