Yes and no. The LTIR player's salary still counts against the cap but the Kings can spend a certain amount over the cap based on a formula. That formula is LTIR Player Cap Hit - [Salary Cap - Total Salary (essentially Available Cap Space)] = LTIR Cap Overage.
I don't know the exact number the day Greene was placed on LTIR and it's already been updated but let's assume Player X goes on LTIR today and he makes $1.8 million cap hit. The Kings have $3.16 million in cap space available as of today with Greene on LTIR.
Salary Cap: $71.4 million
Total Salary: $68.2 million
Available Cap Space: $3.16 million
Player X Cap Hit: $1.8 million
Using the formula, $3.16 million - $1.8 million = $1.36 million LTIR Cap Overage, meaning the Kings could spend a total of $73.2 million on a $71.4 million cap hit and not be penalized while the player is on LTIR.
That $1.36 million would be the amount the Kings could go over the cap while Greene is on LTIR. The Kings (nor any team) would do this because they like to leave buffers, but theoretically the Kings could spend up to $4.96 million ($3.16 million available space + $1.8 LTIR million cap overage) on a replacement player.
If/when Greene is ready to return, moves would need to be made to become cap compliant again.